What is the ruling on vowing to give a sum of money to someone as an apology, with the intention that God may forgive the one making the vow, if it becomes clear that this person will spend the money in a way that harms them and contradicts the religion?

Your statement about giving money to your relative is not a covenant with Allah Almighty. And as long as you do not know whether it was a covenant or not, this means that you did not intend it as a covenant or a vow. So, do not open the door to misgivings for yourself. Do not demand anything of the money, and strive to mend what is between you and your relative.