The answer is yes, because this is not a major irrevocable separation, but a minor one. Thus, he has no right to take her back, though he may enter into a new marriage contract with her.
Know that khulʿ has no irregular form; that is, it is permissible even during menstruation, because it is not a divorce. Allah only commanded that divorce be pronounced with regard to the waiting period: “When you divorce women, divorce them for their waiting period.” [al-Talaq 65:1] Therefore, a man may enter into khulʿ with her even if she is menstruating, and he may do so even if he has had intercourse with her during that state of purity, because it is not a divorce but a ransom. Moreover, the underlying reason a husband is forbidden to divorce during menstruation, or during a state of purity in which he has had intercourse with her, is the harm it causes her by extending her waiting period. If she agrees to that, she has waived her right.
“If it takes place using the wording of khulʿ, annulment, or ransom, and he does not intend it as a divorce, then it is an annulment and does not reduce the number of divorces, …”
His statement, “If it takes place using the wording of khulʿ, annulment, or ransom, and he does not intend it as a divorce, then it is an annulment and does not reduce the number of divorces.”
These are two further scenarios. If it takes place using the wording of khulʿ, annulment, ransom, or similar expressions, and he does not intend it to be a divorce, then it is an annulment. But if he intends it as a divorce, then it is a divorce. Thus, if he says, “I have entered into khulʿ with my wife for one thousand riyals,” it is an annulment; but if he intends divorce, it becomes a divorce. If he says, “I have annulled my marriage to my wife for one thousand riyals,” it is an annulment. And if he says, “I have ransomed my wife for one thousand riyals,” it is an annulment.
Thus, there are three expressions of annulment: khulʿ, annulment, and ransom—provided that he does not intend divorce by them. If he intends divorce by them, then it is a divorce. The correct view, however, is that it is an annulment even if he intends divorce, and even if he explicitly uses the word divorce. Accordingly, there are four scenarios: that it takes place using the explicit wording of divorce; that it takes place through an indirect expression accompanied by his intent; that it