Is it permissible for a husband to ask his wife to waive her mahr (dower) or to divorce her, after he discovered she had an illicit relationship, which she herself described as "fahisha" (an abominable act), based on Allah’s saying: "And do not restrain them [from marriage] in order to take part of what you have given them, unless they commit a clear immorality (fahisha)"?

Your wife has committed a sin that requires repentance. As for you, if she repents and amends her ways, then keep her and admonish her. If she persists in her state, then separation is advised. It is not permissible for you to mistreat her in order to force her into khulʿ (divorce at the wife's instance), even if you are certain of her adultery. If khulʿ occurs due to her adultery or similar forbidden acts, it is not disliked, but you would be sinful for mistreatment. Khulʿ is not valid if you coerced her into it through beating or similar means. You are permitted to ask her to relinquish her mahr (dowry), but it is not permissible to force her to do so. There is no blame upon you for divorcing her, whether through khulʿ or without it, if it is with her consent in the case of redemption.