It is vital that you have documentation to prove that he doesn't pay child support
But he does Terri. The OP said that he pays child support, though he's late with the payments. And he does visit the child, though sporadically. I think it's very unlikely that the court would remove his parental rights. There needs to be a serious neglect of the parental responsibilities for that to happen. The correct term btw is parental authority. The best that's likely to happen is a limiting of his rights.
"If the deprivation of the parental authority is found not to be justified (for example, the father supports the child financially even if it is not on a regular basis and in the amounts smaller than it seems to be appropriate."
The legal position regarding relocation abroad is this:
"When parental authority is granted to both parents, each parent has the right and at the same time is obliged to exercise it (Article 97§1, Family and Guardianship Code (FGC)). Parents should decide together about the important matters in the child's life.[/b]"
The information above which was provided by Dr Joanna Kosińska-Wiercińska and Professor Jacek Wierciński, is available in full here:
I'd say that the chances of the court allowing the child to move would be heavily influenced by the economic circumstances of the OP. Can he afford to send the child back to Poland for regular holidays or pay for the grandparents to come out to the USA and visit. If not, I think it's not likely that the court would allow such a young child to relocate to a country so far away.