you need to contact a local attorney in your jurisdiction to examine the matter for you. Typically, paying a ticket in advance of a court date is tantamount to a guilty plea to the subject offense, so appearing in court for that specific purpose would not appear of benefit to you. The real question here is whether you can file an appeal to the guilty finding already imposed by virtue of your paying the citation. Part of the answer may lie with how much time has passed since the citation was paid because any applicable appeal period may already have run. Contact a local NJ barred attorney in your area .
By paying your ticket you pled guilty. A letter to the judge will not re-open your case. You would need to file a motion to withdraw your guilty plea, which will be difficult. You can get a 2 point reduction by taking a defensive driving course.
The proper procedure is to file a Notice of Motion seeking Post Conviction Relief ["PCR"]. There is a 5 year time limit as to the filing of such PCR motion. The movant needs to state why they are seeking to vacate a guilty plea and the effects of such plea at this time. The movant also need to address whether or not they had an attorney represent them at the time of their guilty plea. You may want to retain an attorney for this PCR motion process.