What should a young man do who was in a relationship with a girl, then learned of its prohibition, and fears breaking her heart if he leaves her, especially since her family became aware of the matter after his father's death? Should he deny the relationship and lie, or admit to it and thus continue the forbidden relationship?

The relationship between a man and a non-mahram (definition) woman is sinful, and it must be severed immediately, and one must repent to Allah. Any heartbreak the woman may experience due to the severing of the relationship is a result of her own negligence and disobedience. This is a whisper from Satan to divert the man from repentance. Lying is forbidden. However, one may use ma'areedh (allusion/equivocation) when asked about the relationship, intending to deny it after repentance.