In Book One in The Odyssey, what problems did Odysseus' absence cause for his wife and son?
The problem that Odysseus's wife (Penelope) and son (Telemachus) are having due to his absence is one of the main bases of the plot. The problem is that there are all these men who want to marry Penelope. They have come in and have pretty much set up camp in Odysseus's hall. By staying there, they are annoying Penelope (bugging her to marry them) and they are using up lots of Odysseus's wealth.
Because of this, Penelope is really mad at them. When Odysseus gets home, he will be even more angry and his actions towards the suitors will be the major event of the last part of the epic.
In addition to the great frustration these suitors have caused Penelope, Telemachus has been growing up in the shadow of a great reputation having no opportunity to learn from this great and mighty warrior, King Odysseus. The suitors make fun of him and he has to watch their ill treatment of his mother. No son wants to experience that.
Telemachus' life will turn out to be a childhood and adolescence of growing fury and anger. Fortunately for him, there will be a time and a place when he will get to display that.