Khula is not regarded as a divorce (talaaq), but it is an annulment of the marriage. The wife cannot go back to the husband after khula except with a new marriage contract.
One of the differences between annulment and divorce is that annulment is not counted as a talaaq, so if you go back to your husband now, he still has the option of three talaaqs.
But if he has given you one talaaq and your iddah has ended, if he does a new marriage contract with you, he only has two talaaqs left.
Any word that is indicative of separation, accompanied by payment of compensation on the part of the wife, is khula.
If the husband utters words of talaaq along with the khula, such as if he says, I divorce you on condition that you return the mahr to me, this is an annulment of the marriage according to the correct opinion. In other words, the khula is an annulment even if the husband utters the word of talaaq at the same time. For more information, please see question no. 126444
You can celebrate the child's birth and do the aqeeqah even if you are separated, and there is no need to repeat the marriage contract for that purpose. But it should be noted that he is now a stranger (non-mahram) to you, with all the rulings that implies.
We advise you to think, pray istikhaarah and examine your former husband's situation; if you think that he is good and that there is some benefit in going back to him, that will be a good time to repeat the marriage contract.
And Allah knows best.
Reproduced from Islam QA