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If you have no custody orders in place and paternity was never established by a court judgment, then you have NO rights, other than the right to establish paternity and custody. If you want to obtain some rights, then you need to petition the court to establish paternity and custody and personally serve the mother while she is in the jurisdiction. You may also be able to obtain an emergency restraining order to prevent the child from being removed from the State of NY and the USA, until your rights are established.
Get a family law attorney immediately, because once the mother and child leave the USA again, you will lose any meaningful ability to establish your rights to a relationship.
That is NOT the same thing as a judgment of paternity from a court. The court must confirm paternity and then make custody and support orders.
My comments remain unchanged.