An analysis of whether you can get full custody and/or supervised visitation involves a detailed assessment of your individual facts. If you were the primary parent to the children, then same is a big factor in your favor. Moreover, domestic violence (which is proven to have taken place) is a specific factor the Court must take into consideration on either the custody issue, or in deciding whether to have his visits supervised versus unsupervised. That said, I highly suggest that you follow-up by scheduling a consultation with a NYC Divorce attorney.
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This is a difficult question.
The more you fight his petition for custody, the more you expose yourself to court scrutiny by way of a custodial evaluator (a licensed psychologist) and a law guardian (attorney for the kids). Hopefully you are a fit parent.
The central issue in your case I see is that you nailed him for domestic violence and he is paying you back with a demand for visitation. You'll have to lay on thick the story about "abuse" and he'll lay on thick your problems with the kids. It sounds like a dance that will lead off the cliff.
We cannot advise you how to foul up his efforts as that would be against our rules of ethics. People have complained about lawyers for years and rules of ethics govern us as never before. It has to be a fair fight with court personnel developing the case and you and he presenting your facts. Many women complain that fathers are getting away with too much in family court but I can say I see women doing very well as well. It is up to a court.