It's not so much that you have no rights, but the question will ultimately be that if she cannot effectively raise the children here in New York - and has a support system in Florida - that will weigh in her favor. If she's threatened to move, then you should file a petition requesting an order preventing the move. Then the question will be whether she busts your chops of the support arrears. That said, you're best advised to schedule a consultation with a Westchester Co. Child Custody attorney.
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Support issues are not determinative of custody issues. You can file a Petition for Custody in New York while the child still lives in New York. If the child moves to Florida then most likely you will have to litigate in Florida.
You would be well-served to discuss your dilemma with a New York domestic relations attorney in a confidential forum as soon as possible.
If you petition to prevent relocation, the mother will have to show how the move is in the best interest of the children. Since you are broke, ask for an assigned attorney may be available, as they are in custody cases. She will have to demonstrate that she has a decent job lined up or some other compelling reason to move thae will benefit the children. You would want to be able to show that you have been exercising reasonable visitation. In short, you do have rights, but you will have to fight for them.
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