Does the woman have a right over the asker in the sight of Allah, knowing that her son breached the agreement, and it is likely that the woman will support her son's statements in the report despite their contradiction of the truth?

If the injured woman is at fault, she has no right against the car owner. If she absolves her of her right, then her conscience is cleared, and she is not obligated to pay anything. If the promise of compensation was made in exchange for not reporting the matter to the police, and they did report it, then they are not obligated to fulfill their promise. If the injured parties do not admit their fault, then the case must be referred to the competent authorities to determine responsibility.