This will not resolve with warm water treatment. Pus signals infection--it sounds like your cat has conjunctivitis. Your cat needs to be examined by a veterinarian to make sure there is not a corneal ulcer present or an ocular foreign body. Your veterinarian can also prescribe topical medication to treat this infection.
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I think your cat may have conjunctivitis, go to the vets to ask for a solution!
Well I think try warm water first wait till it gets older like 2 or 3 weeks and if he/she still has puss around its eye take it to the v.e.t