Does swearing on the Qur’an that one has no relationship with a married woman—given prior attempts at incitement and striving against one’s own soul—constitute falling into a forbidden act that necessitates expiation, or do striving against one’s soul and seeking forgiveness avert that?

The questioner must repent to Allah for the forbidden relationship with the married woman, and for the false oath (yamin (definition) ghamus) sworn upon a lie, and he must cut off his relationship with the woman. The door to repentance is open, but one must be wary of returning to such relationships and of swearing by Allah upon a lie.