Is it permissible to claim rights after the family has requested a divorce, and after having relinquished a part of them?

It is not a condition for khulʿ (divorce at the wife's instance) that the wife forfeits the entire mahr (dowry); rather, they may agree upon a part of it, or more than it, or another compensation. The Malikis and Shafi'is held the view that it is permissible for the husband to take compensation from his wife in return for divorcing her, whether the compensation is equal to what he gave her, or less, or more, and whether it comes from her or from someone else, and whether it is the sadaq (dowry) itself or other wealth, as long as both parties have agreed. Therefore, as long as you have agreed to forfeit a specific amount of the mahr, there is no objection to that.