The nose felt dry, then a bit red in the middle, and where nose meets mouth seems to have blood

Hello. From the photo, the cat has nasal mucosa ulceration. The cause is unclear, mainly considered respiratory problems, calicivirus infection, or a scratch.

Watch for other symptoms; if any appear, take it to a vet for diagnosis. If none appear, it is likely a scratch; clean with saline and disinfect with iodine. To stop licking, put a small bandage or adhesive strip on the area.