Is the husband entitled to reclaim the dowry paid to his second wife who is seeking divorce without cause, and is this considered a khulʿ?

Allah has permitted a man to marry more than one wife. It is not permissible for a wife to seek divorce merely because her husband has married another. If she insists on seeking divorce, the husband may divorce her in exchange for her returning to him what he paid her as mahr (dowry), based on the Almighty’s saying: "And it is not lawful for you to take back anything of what you have given them, except when both fear that they will not be able to observe the limits of Allah. But if you fear that they will not be able to observe the limits of Allah, then there is no blame upon them for what she gives in return for her freedom. These are the limits of Allah, so do not transgress them."