The causes of herpes, both genital (usually HSV-2) and oral (always HSV-1), are viruses. No virus is inhibited by neosporin or any other antibiotics. By definition, antibiotics are active only against bacteria and some fungi and parasites, not viruses.
The moisturizing effect of the other ingredients in Neosporin (or in any other ointment or cream) might make a herpes lesion feel better sooner, but they have no effect whatsoever on speed of healing and do not reduce the amount of virus in the lesion. In any case, no herpes lesion can go away in less than a week and it's usually 10-14 days before healing is complete.
Therefore, the rapid improvement of the sore in your genital area is pretty good evidence against herpes as the cause. Further, a single small cut-like lesion is not typical for the first herpes outbreak, and none of your other symptoms suggest herpes. And you are at low risk for catching genital herpes from your partner anyway. Most people with genital herpes due to HSV-1 do not have frequent recurrences (about half have no recurrent outbreaks after the first one) asymptomatic viral shedding if much less frequent than for HSV-2. I have never seen a case of genital HSV-1 acquired by genital sex; every case I ever heard of was acquired by oral sex. Genital-genital HSV-1 infection probably occurs, but seems to be rare.
So my bet is that you do not have genital herpes. But if more typical herpes symptoms appear, or if the cut-like lesion reappears, to be safe you should immediately see a provider for diagnosis (before putting Neosporin or anything else on it).
I hope this helps. Best wishes-- HHH, MD
Thank you for your help and time! I greatly appreciate it! Best regards.