Please get a biopsy of this. My cat had a similar sore and we waited too long to get a biopsy as the vet didn't think it was anything. Turned out to be squamous cell carcinoma, the skin type. This is a rare place for it and my cat wasn't white and almost always it is white cats that get this and it is on their nose or tip of the ear.
My cat is going to die from this probably soon. Nothing we did cured it unfortunately and we did try chemo, removing it etc. She is maintaining at the moment with a feeding tube and we are waiting to see if the radiation treatments will shrink it and buy her some time. She is 15 years old so that would make a difference. She is ok, at the moment, seems to be enjoying life though in a way less energetic way. Nothing we did as far as treatment bothered her much, the tube doesn't and she had no side effects from the radiation and chemo. I wish we had insisted on a biopsy when it was as small as your cat's because I did see it then. Thought it was a scrape from another cats claw that she'd scratched and it had become an abscess. The vet thought so too.
This is an odd location for this but your cat is white so it makes it less odd that it could be what my cat has. My cat is long haired and dark colored and it is unheard of for a cat like her to get the skin type, sun related cancer. It is in the wrong spot for acne but could be scratching from an allergic reaction to something as that is around the head, but there would not be only one sore. My mother's white cat has sores around her head from scratching