How does Jerry's mom's motivation advance the plot of the story
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.