Did Telemachus have problems growing up without a father in Homer's The Odyssey?
This question cannot probably be answered with a strict yes or no. While it is true that Telemachus spends most of his growing-up years in The Odyssey, by Homer, without a father, things are not all bad for him--though some things are very bad.
First of all, he grows up in a city without a leader (his father) and a home without a father. Just as the suitors have taken over his home, so have others taken over control of the city. It is a disorderly place to grow up.
Second, he has to endure the suitors, watching them literally eat (and drink) up his inheritance. His mother is unhappy but does not feel she can do anything to change things, and neither does Telemachus. Even worse, he kind of whines about it:
"If only the gods would give me such strength as he has to take revenge on the suitors for their overbearing oppression...[but] no, the gods have spun out no such strand of prosperity for me and my father. Now we must even have to endure it."
Finally, he is deeply disappointed that his father, if he had to die, did not do so in battle, as that would have brought glory to his name and therefore to Telemachus, as his son. He says:
I should not have sorrowed so over his dying if he had gone down among his companions in the land of the Trojans, or in the arms of his friends, after he had wound up the fighting. So all the Achaians would have heaped a grave mound over him, and he would have won great fame for himself and his son hereafter. But now ingloriously the stormwinds have caught and carried him away, out of sight, out of knowledge, and he left pain and lamentation to me.
Clearly being lost at sea (which is of course what he thinks happened to Odysseus) is not a glorious, fame-and-glory-inducing death.
On the other hand, Telemachus is the son of a king and has much more than most young boys who are left fatherless. One of the benefits of being Odysseus' son is that he captures the attention of Athena. While it is true that she would not have helped him if he had not also been worthy on his own merits, Athena takes pity on him for being without Odysseus and helps him mature into a fine young man in every way. She tells him:
"Telemachus, you are to be no thoughtless man, no coward, if truly the strong force of your father is instilled in you; such a man he was for accomplishing word and action."
Athena compares son to father and wants Telemachus to be a more "thoughtful" man than his father but recognizes that he needs to learn to accomplish things, as well.
She guides the young man on his journey, and by the end of the story Telemachus is nearly his father's equal, as demonstrated by the fact that he is able to string his father's bow, though Odysseus does not have him do it.
Perhaps Telemachus would have grown up to be a better man under the tutelage and guidance of his father; however, his difficulties of growing up fatherless caused him to grow in stature, particularly after Athena began to mentor him. Being without his father for his growing-up years creates problems as well as possibilities for Telemachus.