What is Jerry's mother's motivation in "Through the Tunnel" and how does it advance the plot?

Jerry's mother does not want to be the smothering type; she is worried that because she is the sole parent she has been "keeping him too close" to her. So, she tries to be loving, but provide him certain freedoms.

When Jerry and his mother walk toward the customary beach one day on their vacation, she notices that Jerry looks over his shoulder, and as he plays on the "safe beach," he thinks of the other one. So, the following day when it is time for their swim, she asks Jerry if he would like to go to another spot. At first, Jerry says "no" because he does not want to hurt her feelings, but then he blurts out that he would like to explore the rocks. So, she gives her permission for Jerry to go to the other beach alone. With this action Jerry's mother provides her son the opportunity to be more independent.

Another way in which Jerry's mother affords him chances for independence is by purchasing swim goggles, and not questioning him about his reason for wanting them. Further, when she talks to Jerry, she says things "casually" without dictatorial tones. On the day that Jerry completes his rite of passage through the tunnel, he returns home and rushes into the bathroom so that his mother will not see the bloodstains or tear stains on his face. After Jerry comes out, his mother just asks him, "Have a nice morning?" and she lays her hand on his shoulder momentarily. Then, she examines him more closely and is concerned, "How did you bang your head?" When Jerry answers "Oh, just banged it," his mother catches herself, thinking,

"Oh, don't fuss! Nothing can happen. He can swim like a fish."

By catching herself, Jerry's mother allows her son to retain his sense of maturity. As a result, he volunteers information on what he has been doing, "Mummy, I can stay under water for two minutes--three minutes, at least." She affirms his accomplishment, "Can you, darling?" But, she also cautions him lightly, "Well, I shouldn't overdo it. I don't think you ought to swim any more today." 

By allowing Jerry the opportunity to make his own decisions, Jerry's mother provides her son room to mature. This freedom advances the plot because she encourages Jerry to make every effort to go through the tunnel and complete his rite of passage. Perhaps, without the trust of his mother, Jerry might not have had the perseverance to have swum through the tunnel.

Describe the initial relationship between Jerry and his mother in "Through the Tunnel".

The relationship between Jerry and his mother at the beginning of the story is pretty typical of children at Jerry's age (eleven) and their parents.  His mother is conflicted about how much freedom she ought to allow him: she doesn't want to smother him but neither is she ready to give him complete independence.  When he expresses a wish to go to the "wild bay" alone, rather than with her to their usual "safe beach," she thinks

Of course he's old enough to be safe without me.  Have I been keeping him too close?  He mustn't feel he ought to be with me.  I must be careful [....].  She was determined to be neither possessive nor lacking in devotion.

Like most parents, she's torn about offering him the freedom he needs in order to become an adult.  She wants to protect him and keep him safe, and this becomes impossible if he achieves independence from her.

Moreover, Jerry feels conflicted about his mother, too.  On the one hand, he really wants his freedom, but he also feels obligated to her, perhaps because of her status as a widow.  He knows that she is alone without him, and so he first declines her offer of freedom, "smiling at her out of that unfailing impulse of contrition -- a sort of chivalry."  He knows he will feel guilty if he leaves her on her own.  He is later "lonely" without her, when he is in his "wild bay," looking at her on her regular beach.  Even by the end of the story, he still very much desires her attention and approval (despite the sense we get from his desire for the older boys' approval at the rock that her attention and approval will soon no longer be enough for him).

In "Through the Tunnel", how has the relationship changed between Jerry and his mother by the end of the story?

I think you would benefit from looking at the story in much wider terms - certainly the Jerry at the end of the story is very different from the Jerry at the beginning, and you are right to identify that one way he has changed is in terms of his relationship with his mother. However, more generally, by the end of the story Jerry has gone through a journey from childhood to manhood, symbolised most stridently in his journey through the tunnel.

At the beginning of the story we are introduced to a character who is on the cusp of adolescense, and very clearly feels responsible for his mother due to their enforced intimacy. Yet despite his feelings of responsibility towards his mother, he nonetheless feels drawn to the "wild beach", which is away from the "safe beach" and his mother's attentive care. The wild beach here can be said to symbolise independence and life away from the protection of a parent figure - note how Lessing describes the two beaches to draw out this comparison.

His discovery of the tunnel and the challenge that the French boys set him through swimming through the tunnel spur Jerry on to train hard and eventually succeed in his attempt to go through the tunnel. Although certainly at the beginning of the story it is Jerry's need to be accepted by the older group of French boys that drives his desire to go through the tunnel, it is interesting that at the end of the story he no longer feels this is the case, as he is happy to go back home and spend time with his mother. This indicates that the tunnel was more about a process of self-acceptance and doing something to show he could do it for himself rather than for any other reason.

His relationship with his mother likewise has changed by the end of the story. Jerry deliberately witholds his triumph, only relating his ability to hold his breath. The dramatic irony in his mother's response ("I wouldn't overdo it, dear") indicates the independence that Jerry has achieved in his journey through the tunnel - he has now entered an arena where he has secrets from his mother and is able to engage in activities, dangerous activies, away from his mother's protection. He is no longer subject to her and has proved himself a man in his own right.

How is Jerry's growth and evolving maturity reflected in his relationship with his mother in "Through the Tunnel"?

When the story begins, Jerry is anxious for some freedom from his mother's watchful and protective eye.  However, "Contrition sent him running after her."  He felt badly for wanting this independence and doesn't leave her on this first day of vacation.  This year, he's much less interested in the "safe beach" they've always frequented in the past; he wants to go to the "wild bay," alone.  

The next day, he gets his chance, and when he looks back at her beach, he feels "relieved at being sure she was there, but all at once very lonely."  He misses her but soon becomes distracted by some older "boys -- men to Jerry" who come along and take turns diving off the rock.  They eventually amuse themselves by swimming through a tunnel in the rock, and since Jerry cannot do it, he begins to clown around to refocus their attention on him.  It doesn't work, and they leave him, crying like a child.  Cried out, he "swam out to where he could see his mother.  Yes, she was still there, a yellow spot under an orange umbrella."  He seems to want to be free of her, but -- at the same time -- he wants to know that she is nearby.

As he begins to grow more confident in the water, he feels that his old beach "now seemed a place for small children, a place where his mother might lie safe in the sun.  It was not his beach."  And next time, when he goes, he does not ask her permission first.  It is at this point that "A curious, most unchildlike persistence, a controlled impatience, made him wait" to attempt swimming through the rock yet.  His ability to delay gratification provides evidence of his growing maturity (he was unable to do so earlier when he accosted her for goggles), as does the fact that it doesn't occur to him to ask her, anymore, if he can go to the bay without her.

In the end, after his experience in the tunnel has seriously frightened him, Jerry does return to her, still a child -- at least, for a while.  He calls her "'Mummy'" and clearly wants her approbation and praise for his new ability to stay underwater for three minutes.  When he tells her this news, "It came bursting out of him."  Jerry has obviously begun the process of maturing, though it seems that one's progress toward maturity is not a straight line, and we can see that in his fluctuating relationship with his mother.

In "Through the Tunnel, "what kind of relationship do Jerry and his mother have?

"Through the Tunnel" opens with a boy and his widowed mother vacationing at a beach resort to which they have come frequently. Because he has no father, Jerry's mother, perhaps, feels more protective of her son than a married woman would, yet she is "determined to be neither possessive nor lacking in devotion." And, while Jerry  feels the loss of his father, he thinks that he must be more attentive to her, acting from an "unfailing impulse of contrition--a sense of guilt about his father's death?--a sort of chivalry." At any rate, they seem devoted to each other in their solitariness at this beach.  For, when Jerry first swims out in the ocean a good ways, he searches for his mother on the beach:

There she was, a speck of yellow under and umbrella that looked like an orange peel.

On the other hand, his mother's devotion is divided between her maternal protectiveness--"Why, darling, would you rather not come with me?"--and her burgeoning understanding that eleven-year-old Jerry needs to be given some independence. For, after he asks her for swim goggles without giving an explanation, she 

gave him a patient, inquisitive look as she said casually, "Well, of course, darling."

Yet, she still remains quite protective because she immediately becomes concerned when Jerry returns after successfully swimming through the tunnel, achieving his "rite of passage" to maturity.  "How did you bang your head?" she asks, and when he dismisses this injury, Jerry's mother becomes "worried." However, the mother does begin to recognize Jerry's maturation as she says to herself, "Oh, don't fuss! Nothing can happen. He can swim like a fish." Still, she cautions him not to "overdo it." This recognition of the mother is reinforced by Jerry's feeling that his mother's warning is "no longer of the least importance." Clearly, their relationship has grown from protective mother and child to an understanding mother and maturing son.

What motivates Jerry's mother in "Through the Tunnel" and how does this advance the plot?

Jerry's mother is motivated by her concern that she give Jerry enough freedom, freedom appropriate to a quickly-maturing eleven-year-old boy, but not too much, as too much freedom might be dangerous. "She was determined to be neither possessive nor lacking in devotion." She doesn't want to smother him with her attention, but she also wants to protect him from anything that could be harmful to him. Further, she doesn't want Jerry to think that she doesn't care about him.

Her motivations help to drive the story because, were it not for her scrupulous attention to his behaviors, Jerry would likely not have even gone to the bay. He doesn't ask her at first if he can go because he doesn't want to hurt her feelings (he is keenly aware of her concerns); it is she who notices his thoughtful gaze toward the "wild bay," and she who offers him the opportunity to go there. Further, it is her desire to grant him independence that enables him to go day after day, even when she notices that he looks like he's "overdoing" it. Without her fairly typical concern and her recognition of his age-appropriate desire to have some independence, Jerry would never have gone to the wild bay and witnessed the older boys swimming through the tunnel.

How does Jerry's relationship with his mother reflect his maturity in "Through the Tunnel"?

In the beginning of the narrative of "Through the Tunnel," Jerry stays close to his mother, asking permission to part from her, then returning to her after going to the rocky bay; later, he demands goggles and goes independently to the rocky bay.

• Dependence upon his mother

Because his mother is a widow and Jerry an only child, she is protective of her son; likewise, Jerry feels emotionally tied to her. At the beach, when she feels that he is not with her, she quickly turns around,

"Oh, there you are, Jerry!" ....Contrition sent him running back to her, And yet, as he ran, he looked back over his shoulder at the wild bay; and all morning...he was thinking of it." 

As Jerry's yearning to explore the rocky bay waxes, his mother senses that she may be keeping him too close to her now that he is eleven years old. So, although she is anxious, she gives Jerry permission to go the big beach. Jerry is excited to go, but still turns to find his mother on the beach:

When he was so far out...he floated on the buoyant surface and looked for his mother.

• A burgeoning independence

After his experiences with the boys who swim under the water and rebuff him, Jerry swims back to the other beach and returns to the villa where he and his mother are staying. He waits for her, and as soon as his mother enters, Jerry demands swimming goggles, "pant[ing], defiant, and beseeching." She agrees, but he "nagged and pestered" her until she takes him to a shop. Then, Jerry grabs the goggles and runs off without asking permission.

As quickly as he can, Jerry swims to the big barrier rock. Adjusting his goggles on his face, Jerry dives into the deep water. However, the force of the dive knocks his goggles out of place; so, he fills his lungs and adjusts his goggles in order to see below.

Jerry explores the rock and finds the hole. Then, he practices and practices until he feels dizzy. But, when his nose bleeds that night, his mother cautions him and insists that he accompany her the next day.

It was a torment to him to waste a day of his careful self-training, but he stayed with her...[where it] now seemed a place for small children....It was not his beach.

• Independence from his mother and a new maturity

The next day, Jerry does not ask permission to go to "his beach." Instead, he leaves before his mother has time to consider the "complicated rights and wrongs of the matter." When he arrives at the wild bay, Jerry ponders trying to go through the tunnel, but "[A] curious, most unchildlike persistence, a controlled impatience" leads him to wait.

After his mother tells Jerry that they must return home in four days, Jerry realizes that he must attempt to swim through the tunnel "if it killed him." So, he dives down, controlling himself, but feeling panic along with a sense of victory that he is moving through the tunnel. He knows that he must continue or he will drown. Finally, Jerry reaches the surface, although he feels that he may not be able to swim back to the rock. Yet, he manages, and when he removes his goggles, they are filled with blood.

In a little while, Jerry returns to the shore and makes his way up the path to the villa. He rushes to the bathroom to wash away the blood and the tear stains, but his mother senses that Jerry is too pale under his suntan. "How did you bang your head?" she asks him, while at the same time warning herself not to worry. Jerry only tells her that he can remain under water for three minutes, at least. Moreover, "it was no longer of the least importance to go to the bay" as he has already completed his rite of passage. 

Why does Jerry yield to his mother's demand at the end of "Through the Tunnel"?

The final words in the story best summarise why Jerry did not challenge his mother's instruction that he should not go swimming any more that day:

It was no longer of the least importance to go to the bay.

The reason for this is that he had just gone through an extremely difficult trial that he had put himself through. He had almost lost his life and had to overcome his fear and tremendous difficulties to complete the arduous task that he had set himself.

The reason why Jerry had imposed this amazingly strenuous task on himself was that he had to regain the dignity he lost when a number of older boys humiliated him when they swam away from him during his visit to a wild bay. They had been diving into the water and swimming through an underwater tunnel running through a rock.

Jerry's attempts at diving were quite pathetic because the salt water stung his eyes. He could see the boys' disapproval when he tried to prove his worth by performing some foolish tricks and he felt ashamed. His attempt at speaking their language was also ignored and they later left him alone. Their rejection so humiliated him that he cried copiously.

It became imperative for Jerry to prove himself the boys' equal and he asked his mother to buy him a pair of swimming goggles. He returned to the rocks and persistently tried swimming through the tunnel. Two days before the end of their vacation, he decided that he had to succeed on that day. There would be no other. In the process, he had to overcome tremendous obstacles: he was afraid, his nose had been bleeding and the tunnel was narrow, but he refused to give up.

Jerry had taken an amazing risk, but his persistence paid off. He finally got through the tunnel, and almost lost consciousness during his struggle. When he finally got through, his eyes and nose were bleeding, but he was generally fine. His achievement was a source of tremendous pride for him. He had proven that he was the boys' equal and that their rejection had been an oversight.

With his self-respect restored, Jerry had nothing more to prove and going back to the beach to swim was, therefore, not an issue any more. 

Why does Jerry yield to his mother's demand at the end of "Through the Tunnel"?

In Doris Lessing's "Through the Tunnel" the conflicting forces of the authority of Jerry's mother and her son's freedom move the plot throughout the story. However, after Jerry practices holding his breath and finally succeeds in swimming through the underwater tunnel he completes his rite of passage and feels the equal of the older boys who have shunned him earlier. Consequently,

[H]e did not want them.  He wanted nothing but to get back home and lie down.

When Jerry returns home, he is simply ready to rest after his exhausting venture. But, he does want to inform his mother of his accomplishment; and, when she urges him to not overdo his swimming challenges, she

was ready for a battle of wills, but he gave in at once. It was no longer of the least importance to go to the bay.

Jerry no long has anything to prove because he knows that he is capable of doing whatever the older boys do, and he feels he has matured now and is no longer a young boy. Therefore, he acquiesces to his mother's wishes which do not carry the importance he once attached to them.   

How can I explain that Jerry has matured throughout "Through the Tunnel" by Dorris Lessing? 

At the beginning of the story, Jerry is repeatedly referred to as "the English boy" and is most concerned with fitting in with the older boys at the rocky bay.  "They were big boys -- men to Jerry" and "To be with them, of them, was a craving that filled his whole body."  Describing him as the English boy emphasizes his youth and immaturity, and his desire to fit in with the older kids also shows just how immature he is; they are children too, and yet Jerry sees them as "men" because of his even more youthful perspective. 

They eventually abandon Jerry, and he runs to his mother to ask her for goggles so that he can attempt the feat that they did, swimming through the hole in the rock.  He wants them "now, now, now!  He must have them this minute and no other time."  Jerry, like an immature child, is incredibly impatient and cannot delay gratification at all.  It doesn't take long, however, for this to change.  After several days of practicing at holding his breath and swimming, Jerry now looks back at the old beach he used to visit with his mother as "a place for small children [...].  It was not his beach."  Now, he does "not ask for permission" to go to his rocky bay; only children ask permission, and Jerry is beginning to grow up.  Further, although he thinks he could probably make it through the tunnel now, "A curious, most unchildlike persistence, a controlled impatience, made him wait."  Jerry has developed some patience, some ability to look ahead and wait for what he wants.  This is quite a departure from his desperation for goggles of a few days ago.

In the end, when his mother tells him not to swim any more today, "She was ready for a battle of wills," indicating that this is what she would typically expect from young, immature Jerry.  However, "he gave in at once" to her and didn't fight her, indicating that he's developed a little more ability to reason and lost some of his former stubbornness (both associated with growing up and becoming more mature). 

Describe the relationship between Jerry and his mother in "Through the Tunnel".

"Through the Tunnel" is a rite of passage story; therefore, Jerry begins in an immature state and reaches a level of maturity which he has not had before.

• Without a husband, Jerry's mother probably feels more protective of her son than a married woman would, yet she is "determined to be neither possessive nor lacking in devotion."
• While Jerry senses the loss of his father, he feels he should be more attentive to his mother, acting from an "unfailing impulse of contrition--a sense of guilt about his father's death?--a sort of chivalry."
• Jerry and his mother seem devoted to each other in their solitariness at the beach. For example, when Jerry first swims out in the ocean a ways, he searches for his mother on the beach.
• When he swims out and tries to interact with the native boys, Jerry goes farther from his mother, still, the boys perceive him as childish when he splashes and acts silly to get their attention. Jerry is perplexed because when he acts this way before his mother, she usually laughs.
• He checks to see that his mother is yet on the beach. Moreover, he swims back to her.
• But, when he swims to the shore, he is insistent that he needs swim goggles. He is beginning to assert himself.
• Unbeknownst to his mother, Jerry returns to the large barrier rock where the older boys have swum and, wearing the goggles, he searches for the gap through which they have passed earlier.
• He discovers a gap and panic fills him, so he swims to shore. But, he plans his strategy for passing the through "that cave" later.
• So, he practices holding his breath, and for the next two days, he exercises his lungs to the point that his nose bleeds at night.
• On the third day, Jerry does not what to accompany his mother to what he now considers a beach for children. "It was not his beach." He is maturing.
• On the fourth day, Jerry does not ask his mother for permission to go to what he considers "his beach." Instead, he leaves early before his mother could think about the risks he might be taking.
• Jerry watches the boys go through the tunnel and counts to 160.

A curious, most unchildlike persistence, a controlled impatience made him wait.

• Jerry goes underwater and checks the tunnel; later, he sits by the clock in the villa, holding his breath repeatedly until he calculates that he must hold his breath for two minutes.
• When Jerry's mother tells him they will return home in four days, he vows that he will get through the tunnel, even if it kills him. Clearly, he feels he must prove that he is no longer a little boy. 
• With only two days left before their departure, and despite a nosebleed the day before, Jerry goes secretly to the tunnel. Although nervous, he puts on his goggles and dives down.
• Despite the pulsing in his head, Jerry perseveres and makes it through the tunnel. "Victory filled him" as he knows that he must persevere or he will drown; Jerry thinks very clearly and maturely.
• When Jerry surfaces and sees the other boys. "He did not want them" because now Jerry feels himself the equal of the older boys.
• Once he makes his way home, Jerry responds very nonchalantly to his mother's inquiries, making light of the knock on his head.
• Jerry's mother looks at him and his concerned, but realizing that Jerry is different now, she tells herself, "Oh, don't fuss."
• Jerry casually informs his mother that he now can stay under water for two to three minutes, and she tries to sound casual, "Can you darling?" But, she still tries to coddle him some, saying he "shouldn't overdo it." 
• However, Jerry does not argue as he would have done. "It was no longer of the least importance to go to the bay." He knows now that he is a young man; he has completed his right of passage.

What changes in Jerry does his mother perceive during their beach vacation in "Through the Tunnel"?

In "Through the Tunnel," Jerry's mother perceives that Jerry wants to become more independent, be more proficient at swimming like the older boys, and attain his rite of passage.

When Jerry and his mother come to the usual beach for the vacationers on their first day of vacation, she notices Jerry look toward a rocky bay and then back at the beach on which they have sat on previous vacations. She asks him if he would rather go somewhere else, but Jerry runs after his mother as though out of contrition for his desires.

And yet, as he ran, he looked back over his shoulder at the wild bay; and all morning, as he played on the safe beach, he was thinking of it. 

On the second day, Jerry's mother asks him again if he would rather go somewhere else. Jerry tells her he would like to have a look at the rocks over at the rather wild-looking bay. After some hesitation, his mother tells him,

Of course, Jerry. When you’ve had enough, come to the big beach. Or just go straight back to the villa, if you like.

Happy she gave him her approval, Jerry hurries to the wild beach and runs into the water. Initially, he feels lonely and looks back at his mother. Soon, however, Jerry ventures out to where some older boys were on some wild-looking rocks. They dive from a point into the sea that forms a pool among the rocks; then they emerge and swim around, pull themselves out, and wait to dive again in turn. Fascinated, Jerry watches. He then swims up to the rock and takes his place to dive, proud he can perform as well as the others.

When the boys dive down under the water and re-appear some distance away, Jerry realizes they must be passing through something under the water. He submerges himself, but cannot see exactly where they swim. He calls out, but the other boys gather their things and depart. Alone now, Jerry returns to the villa where he and his mother are staying. He goes to his mother and demands some swim goggles. As soon as his mother buys him goggles, Jerry runs off to the bay with them in his hands.

Jerry puts on the goggles and immediately submerges himself in his effort to discover the opening in the rocks where the boys have passed. After some time, Jerry discovers the hole of the tunnel through which the others have swum.

He knew he must find his way through that cave, or hole, or tunnel, and out the other side.

Jerry returns to the villa as he realizes he must learn to hold his breath for some time. Also, he must be able to propel himself through this tunnel as a rite of passage to adulthood. He practices holding his breath for hours. He looks at the clock one day after holding his breath and realizes he has held it for over two minutes.

When his mother announces that in another four days they will return home, Jerry decides to attempt his swim through the tunnel. After he submerges himself, he dives inside the hole in the rock. He swims for a while, then worries he will not succeed.

He must go on into the blackness ahead, or he would drown. His head was swelling, his lungs cracking. . . he feebly clutched at rocks in the dark, pulling himself forward, leaving the brief space of sunlit water behind. He felt he was dying.  

Finally, Jerry sees light and swims out through the tunnel, his rite of passage complete. Although his goggles are filled with blood, Jerry is satisfied because he has done what the other boys have. He returns to the villa, where he tells his mother he can hold his breath for two or three minutes.

In "Through the Tunnel" what is the relationship between Jerry and his mother?

We are presented with a very awkward relationship between the protagonist of this excellent short story and his mother as Jerry is on the threshold of adulthood and his mother struggles to work out how best to give him limited independence whilst trying to ensure his safety. Note how we see a conflict between them right at the beginning of the story when she spots Jerry looking at the "wild beach," and how "contrition" forced him to follow her to the "safe beach." Perhaps the most revealing paragraph we have that addresses their relationship comes when the author tells us some background information about the mother and Jerry:

He was an only child, eleven years old. She was a widow. She was determined to be neither possessive nor lacking in devotion. She went worrying off to her beach.

Thus we can see that the mother, as a single mother, is desperately trying to raise Jerry well by herself, and wants to try and get that balance between giving him independence whilst at the same time letting Jerry know that he is deeply loved. Thus it is that she lets him go to the "wild beach" by himself.

In "Through The Tunnel", what role does Jerry's mother play in the theme of rites of passage?

In Doris Lessing's "Through the Tunnel," the tension between the mother's protectiveness and authority and Jerry's desire for freedom cuts through the entire narrative.  A widow who must struggle to provide for her boy, Jerry's mother suffers from the anxieties of having to raise a child without a father, worrying that she may be doing too much:

Have I been keeping him too close?  He mustn't feel he ought to be with me.  I must be careful.

These anxieties of the mother and her consequent reactive protection, ironically, cause Jerry to wish to prove all the more that he is a man by asserting his independence.

When the older boys who swim through the tunnel reject Jerry for his childishness, Jerry becomes aware that he is too protected by his mother who has him sit on the "safe beach." He, then,

knew he must find his way through that cave, or hole, or tunnel, and out on the other side.

The tunnel becomes, of course, symbolic of Jerry's rite of passage from childhood to adulthood. Without his mother's knowledge, Jerry submerges himself in the tunnel and swims through it, although he almost does not make it without exploding his lungs.  But, having made it safely, only suffering a bloody nose, Jerry returns to the villa.  Now, he sees the local boys diving, but "he did not want them" because he knows that he is now their equal.  Once in his room, Jerry rests until he hears his mother.  As they have lunch, Jerry boasts only that he can hold his breath three minutes.  But, secretly, he knows that he has proved himself a man.

How does Jerry's relationship with his mother affect his desire to swim through the tunnel?

"Through the Tunnel" is a great coming of age story. It's great because of how relatable the story is to most readers. Jerry isn't going through a coming-of-age adventure that is pure fantasy. His coming-of-age adventure takes place while on a beach vacation and involves a personal challenge. Jerry wants to swim through a lengthy underwater tunnel. It is a dangerous challenge, because Jerry could lose his life in the attempt. His desire to do the swim and prove to himself that he can train for it and accomplish the swim is complicated by his relationship with his mother. This is an interesting complication, because Jerry's mother isn't against the swim. In fact, she doesn't even know what Jerry is trying to do. The complication exists in Jerry's mind.

Jerry hasn't spent much time away from his mother. He wants to, but the independence is a new thing to him. It feels foreign, and we see this when he goes off to swim alone the first time. Jerry keeps checking back to make sure his mother is still there. He feels obligated to be her kid, stay near her, keep her company, and do as she wishes. That stands in conflict with his need to do some dangerous training away from her. Jerry knows that she would be against the idea, so Jerry has to essentially lie to his mother. This is a hard thing for him to do, but his desire to swim through the tunnel is a stronger pull than the pull to take it easy and stay near his mother.

How does Jerry's mother's motivation advance the plot of "Through the Tunnel"?

I'm not sure I agree with the statement that Jerry's mom's motivation advances the plot of the story.  Jerry has a lot of intrinsic motivation.  He's on vacation with his mom, and he wants to gain a little bit of independence from her, so he asks her if he can go to a different section of the beach/bay.  His mom doesn't fight him on this.  She tries to act casual about it even.  Once there, Jerry sees some "native" boys.  He desperately wants to be a part of their "cool" group.  Unfortunately he can't make the swim through the underwater tunnel.  He feels devastated, but he vows to train hard in order to make it.  He demands goggles from his mom and she agrees to buy them.  He trains hard enough to have nose bleeds, and mom's only reaction is to tell him to take it easy and rest on the safe beach for a few days.  

Mom cares deeply for Jerry, but I just don't see her being a "hovering parent."  If she were, it would make sense how she might motivate Jerry to go off on his own and do dangerous stuff.  But she isn't oppressive to Jerry.  I believe that Jerry self motivates himself to grow up a bit.  To be bigger, stronger, faster through training.  That way he can step toward manhood (in his opinion).  

There is one specific instance where I feel Jerry's mom motivates Jerry to try for the tunnel.  That is when she tells Jerry that they will be leaving to go home in 4 days.  That puts Jerry in a "now or never" situation.  Perhaps Jerry's mother normally is a hovering parent and Jerry has felt very confined and stifled at home.  The vacation might be her first step in turning over a new leaf and letting Jerry have some independence.  Jerry in turn, because of the inch that his mother has given him, decides to try and take a mile.  That would be how Jerry's mother's motivation drives the plot of the story. 

In "Through the Tunnel" by Dorris Lessing, how is Jerry's growth and evolving maturity reflected in his relationship with his mother?

Jerry is eleven years old, and is the child of a widowed mother. He is respectful of his mother and does not to want to worry her in any way. For instance, even though he’d rather spend his time on the “wild and rocky bay,” he stays with his mother in the safe bay. It seems that the two have a close relationship, as his mother is discerning of his thoughts. She notes his disinterest in the safe beach and, not wanting to be overly possessive, lets him venture out to the wild-looking bay when he asks her permission to do so. She thinks that Jerry is “old enough to be safe without her”; this is proof that she would like her son to develop independence in his life. However, she still worries about the best kind of upbringing to give her son—she is “determined to be neither possessive nor lacking in devotion."

Jerry loves his mother and feels the need to always be around her. For instance, when she allows him to go by himself to the wild-looking beach, he “almost runs after her again feeling it unbearable that she should go off by herself.” However, at eleven years of age, he is at a stage in life where he is transitioning into adulthood—hence, his need for independence and his search for self-identity.

At the “wild-looking bay” he meets a group of older boys and yearns to be a part of them. When they accept him into the group, he “is happy to be with them.” Note that, after a long time spent on the wild-looking beach, Jerry swims out to a place from which he can see his mother. Her presence offers his some sort of reassurance. After time spent with the older boys in the wild beach, he resolves to learn to swim through the tunnel in the water, just like they did. After some practice, he finally makes it and shares the exciting news with his mother.

Why is Jerry described as "defiant and beseeching" towards his mother in "Through the Tunnel"?

The narrator of "Through the Tunnel" describes Jerry as both "defiant and beseeching" when he accosts his mother for a pair of goggles.  He requires the goggles in order to do what the "big boys" had done and swim through a tunnel in the rock promontory at the "wild bay."  Initially, Jerry really wants to be mature, to gain independence from his mother, and to be like the big boys who were like "men to Jerry."  However, he is still very much a child, and this accounts for the description of his request as "defiant and beseeching."  He is somewhat rude in his address to his mother because he is only able to think of his one immediate goal.  He needs her to buy him the goggles, but he is too fixated on his goal to bother with the courtesy he showed to her earlier in the story.

In "Through the Tunnel," how does Jerry's mother hinder his initiation?

I believe that the "initiation" that your question mentions is Jerry's initiation into adulthood.  Or in becoming a man, whichever you prefer.  I don't feel that Jerry's mom is too much of an obstacle.  To her credit, she lets Jerry go off on his own on most of the days.  She absolutely could have forced Jerry to stay at her beach within sight of her.  She did do that for one day, which was a minor obstacle, because it prevented him from further attempting the tunnel swim.  She also is an obstacle in the same sense that a lot of moms are to young teenage boys.  Mom, and being seen with mom, simply is not "cool."  Jerry wants to be cool and independent, and he can't do that if he is forced to hang out with his mother.