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If you're seeing the "third eyelid" coming up over the eye (this is a pink or fleshy-colored lower eyelid, there is no fur on it), then this can be a sign that there's a painful problem within the eye. This can range from an injury to the cornea (the clear layer of the front of the eye), or uveitis (inflammation inside the eye), or a nerve problem causing the eyeball to be retracted into the socket.
The third eyelid can become "stuck" to the cornea, but it's not common. This is seen in cases of severe damage to the cornea, often associated with upper respiratory infections such as Herpes virus.
Oral clavamox is probably not going to solve this problem, I would suspect that eye drops or ointments will be needed, to reduce inflammation, treat infection if present, and provide lubrication. It may be a good idea to see a veterinary eye specialist.