A facial boil that burst last week after taking antibiotics. The boil left a small hole and I am wondering if the core is gone o should I continue to clean with hydrogen peroxide solution and neosporin or what?

Will heal. It should heal without any further treatment. The small hole will eventually shrink and smooth out over time. It is possible a small scar will remain but this too should decrease w time.
I . I often have patients take care of wounds like this with diluted peroxide (mixed 50/50 with water) followed by plain vaseline until the wound has healed. Without seeing your face it is hard to say whether you should continue with your regimen or not. I would follow up with whomever prescribed the antibiotic or your regular provider to get it checked out.