Fingernail infection??

So yesterday after I picked my 3 month old up from daycare I noticed that his middle finger on his left hand was red. Once we got home I noticed it was where his fingernail had been sharp on the corner and I guess scratched or dig into his skin. It had a small white place where it had done so. I clipped the nail and filed it and leaned the area well with warm water. kept Neosporin on the spot with a mitten over it since he likes to eat his hands. This morning it still looks like this. It doesn’t seem to hurt him when I touch it. It’s only on his skin, not under or on the nail. Any advice??