The stinging is called envenomation . Some just bite and tear.
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The goal of treatment is to stop bleeding, prevent infection,
and alleviate envenomation, or exposure to poison.
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It's estimated that a few people die from stonefish envenomation each year. Stonefish venom is extremely potent and can be fatal if not treated promptly.
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You should probably take your cat to the veterinarian for an
examination. Facial swelling can be a sign of an intoxication or
envenomation (such as wasp sting or snake bite), or it could be a
severe allergic reaction.