I do not believe that that there is any mechanism for you to renew your passport if you owe $50,000 in child support arrears. The records should show that your passport was suspended a long time ago. You would have to contact the Support Collection Unit to see if they would agree to a payment plan. However, without court intervention, it is doubtful that any payment plan will allow the renewal of your passport. Your only other option would be to contact the mother of your child to see if she would be willing to come to an arrangement concerning the arrears. This is also difficult since as a general rule, arrears are not supposed to be reduced. The court in New York may allow for a reduction in the arrears if you can prove that your income was below the self-reserve amount. You need to consult with an attorney to explore your options.
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It is unlikely that your passport will be renewed unless your case falls between the cracks. If you are a citizen of another country you should get a passport from that country and thus avoid the problem. As for the future, you would have to pay off your arrears or wait until your child is at least 41 for the arrears to go away.
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This advice is not meant to create an attorney-client relationship and is a general anwer to the question posed.