Is the marriage of a wife, after Khul', to a man she had a relationship with before her 'iddah expired, Islamically valid? And is the husband entitled to reclaim the mahr that she waived in exchange for the Khul'?

If a woman marries after her 'iddah (waiting period) has ended, her marriage is valid if it fulfills its pillars and conditions. Whatever she waived from the dowry in exchange for khul' (divorce initiated by the wife) is a right of the husband, based on Allah’s saying: "then there is no blame upon them for what she gives to free herself thereby" (Quran 2:229). It must be noted that it is not permissible for a woman to go out with a non-mahram man during her 'iddah. This impermissibility is further emphasized if the 'iddah is for preserving the husband's right and safeguarding his lineage. She is obligated to reside in the place designated by the husband for her, provided it is safe and suitable.