Is it permissible for the husband to pay off his wife's debt to him from his Zakat money, by waiving the remaining amount for her, or by giving her the money for her to pay him back?

It is not permissible to give zakat (definition) to the wife to pay off her debt if she is able to pay. However, if she is in straitened circumstances, it is permissible to give her from zakat to pay off her debt without any condition or collusion. It is not valid to waive a debt owed by a debtor with the intention of it being zakat; for zakat is giving and transferring ownership, not waiving and absolving, and this is the view of the majority of jurists. Granting an extension to the one in straitened circumstances and giving charity to him is better and greater in reward.