How should one act in this situation, where she confessed to him a past that includes an illicit relationship with her teacher years ago, and now the young man cannot overcome the matter? Should she leave him, or try to mend things with him?

The young man has no right to demand knowledge of your past sin, and you must conceal yourself. If he insists on asking, you can use allusive speech (tawriya), and it should be sufficient for him to observe your current state. If you have repented and become righteous, then he should marry you, for repentance erases what came before it. If the young man is not content to marry you, then supplicate, and ask Allah for a righteous husband, for He may facilitate someone better for you. It is your right to seek a righteous man and offer yourself to him. We advise you to strive to mend your relationship with your parents by honoring them and treating them kindly.