How can one get rid of feelings of lust towards a close friend who trusts you, while avoiding their negative influence that distracts from prayer?

If you are speaking with this young man about sexual desires, then it is impermissible. If you are infatuated with him, this could lead you to fall into immorality, especially given his viewing of movies, listening to songs, and distracting you from prayer. Therefore, you must avoid him to safeguard your religion and your honor.