Lucy: "Mr. Smith is refusing to accept my science project because he says my paper isn't long enough! I just don't
understand; I worked so hard on that and now it was all a waste of time."
Peter: "Don't worry, you can always enter again next year."
Lucy: "I don't know why I bother talking to you, you don't listen to a word I say!"
What adjustments can Lucy and Peter each make to their communication, in order to avoid misunderstandings like this
in the future?

Based on the information that have been given in the question, the adjustments that can be made by both Lucy and Peter to their communication, so that misunderstandings like this will be avoided in the future will be for Lucy to tell Peter to listen to her, support her and show empathy regarding her situation.

Peter, on the other hand doesn't necessarily have to offer her an advice but at least he should show empathy towards her and show that he's concerned.