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The chin is a common place for feline acne but I can't be sure that it's what you're seeing. If you suspect this, you can use an over the counter acne medication on a small area and see if it helps. You will also want to change from metal bowls to plastic bowls. If you are already using plastic, be sure to wash them well in the dishwasher to remove all the bacteria.
Your vet might want to give steroids to help with the itch and give the place time to heal.
If you can get in sooner, it would be a good idea. Most vets have weekend hours now.
Let me know if you still have questions.
Sorry that my answer was not helpful. I can only tell you things that I have experienced. I have used acne cream with cats, and in my experience, it has worked. It might not be the same for you.
BUT, I re-read your description and it does not sound like an acne problem. I would still follow the suggestions about the bowls.
My only other suggestion might be that it's an infection or inflammation of the salivary gland.
Does this help more?
Or do you still have questions?