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It sounds most likely that your cat has ulcers on the eye with conjunctivitis occuring, not uncommon.
The vet should do a pressure test first to make sure there is no glaucoma, either primary or secondary to whatever is going on. Then, they will stain the eye to look at the ulcer(s) and see how deep they are with their light. Treatment is using a triple antibiotic opthalmic ointment with hydrocortisone added, a prescription medication. Due to the hydrocortisone, it cannot be found without a prescription. An antibiotic ointment only would help clear the conjunctivitis for now, but the corneal ulcer(s) will continue to get worse and cause permanent damage to the eye eventually.
Ulcers can be an auto-immune problem, an age problem, a lack of tears problem, or of unknown cause. Your vet may also want to do a tear test, to make sure the eye is making adequate tears, and if not, artificial tears may be added to the regimen of medications to give.
Hope your cat's eye is healed up quick, and please let me know if I can help further.
Corneal ulcers link: