The incident of talaq or khula takes place in case of difference between spouses. In khula, wife withdraws from the right of mahr or compensates some amount to the husband and the husband takes her out of his nikah with the words of talaq or khula. By khula, talaq bai’n takes place. Hence, if the husband and wife want to live together after khula, they can do so after performing new nikah.
Allah (Subhana Wa Ta’ala) knows Best
Darul Uloom Deoband