First, if you decide to petition the court to relocate, the court must determine, based on all of the proof, whether you have established, by the preponderance of the evidence, that a proposed relocation would serve the child’s best interests. This basically means that moving to a new state would be beneficial for the child’s growth. If the move is part of providing the child with a better upbringing, from a financial perspective (i.e. better job, etc.), this is also a very relevant factor.
An additional factor to consider is whether the father would oppose such a relocation request. If the father is not very involved, perhaps he will not show up to court. If he does not show up to court, then you could have a relocation plan agreed without any acrimony.
Based on your question, however, it does appear the father would be against the proposal. If this is the case, the father would have to show why relocation does not serve the child’s best interest. In light of the fact that the father is not very involved in the child’s life, it may be difficult for him to make this argument. However, he could offer facts that are unrelated to his lack of involvement. For instance, if he believes the current environment provides greater stability, he could emphasize that point. However, if the father is not involved, the persuasive force of his arguments may be diminished.
Finally, when the custodial parent does move with the child to another state without the consent of the court or the non-custodial parent, it is very difficult for the court in New York to enforce the order. Once the custodial parent leaves, the legal proceedings can become very tricky and it can be very difficult to enforce the original order.
If you are facing a similar situation, please consult an attorney for legal advice. This is simply a very basic overview of some of the issues but I would need more facts to provide you with any kind of legal guidance going forward.