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I am sorry to hear about Mitten having this problem. My name is Rebecca. I am a veterinarian and will do my best to help.
This could be from an infected tooth, that is a common reason for one side of the face to swell up and then drain. Does he go out, or come in contact with other cats? It could be an abscess from a bite or fight wound.
If this is an infected tooth the swelling may keep coming back. Taking him to a veterinarian to have his teeth checked and getting antibiotics if this is an infection might be the best thing to do for him. It is good that he is eating and drinking so it is not an emergency right now, but I would get it checked out especially if the swelling comes back.
Let me know how he is doing and what other questions you have. Rebecca
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