In New York State, you or your attorney, in court, could request an “adjournment.” You could also request an adjournment for the purpose of securing representation. Whether the court grants the adjournment is within its discretion and the court can consider many factors in making that determination. One consideration will certainly be whether this case has been adjourned in the past. Another critical factor is that the adjournment request is made a reasonable time before the court date, so the court can reschedule the court date.
Another, less formal, option is that you could write a letter to the judge who is deciding the case and ask to have the case rescheduled, or you could write a letter to the court clerk.
Again, there are many factors involved and I do not know the facts here enough to give you reliable advice. One of the most important factors is the court where the case is decided. Different courts have a different process for deciding to adjourn a court date. Some courts require a request be made in writing while others deem an oral request to be sufficient. To find out the process in your case, give the court clerk a call and ask the best way to request an adjournment.