I am sorry to hear that you are dissatisfied with your assigned counsel. I am sure that counsel is over worked and under paid and your case may be routine and not require excessive work in preparation for a hearing. You need a good attorney who has stayed with you on your case to "make a record" which is the process of making motions, getting orders, getting disclosure and running a trial with sound evidence and good objections practice.
You can appeal the custody order to your local Appellate Division. It is an expensive process and can take years. The general idea is to show that the judge's order is so far removed from the record (lacks a sound and substantial basis in the record) that it must be reversed. You start with a notice of appeal, then get calendared while you gather transcripts, prepare and file your brief and then go to oral argument and ask the court to submit your appeal for decision.
Your other option is to finish this case and later refile. You can refile any time in family court as no family court decision is res judicata and all dismissals are without prejudice. I hope you know what you are doing and get a good attorney to help you.
If there is a mistake of law, then you can appeal. You can also file a petition in the future if you qualify. Speak with your lawyer to plan and prepare for the trial.
If this answer is helpful, then please mark the helpful button. If this is the best answer, then please indicate it. Thanks. For further information you should see an attorney and discuss the matter completely. If you are in the New York City area, then you can reach me during normal business hours at 718 329 9500 or www.mynewyorkcitylawyer.com.
You can file an appeal from the decision of the Court if you believe it has been arrived at in error. In order to preserve your right to appeal, you would need need to file a Notice of Appeal within 30 days from the date of the decision.