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Can you gently look at the color under the lower eye lids of both eyes? Does the color look the same in both eyes?
Is the eye with the white drainage sensitive to light or squinting?
It sounds like she may have a lack of tear production in that eye. The eye responds by becomming irritated and producing thick sticky mucous. To diagnose it, he would need to go to the vet and have a tear test performed. It is a simple proceedure. If he does have lack of tear production then there is a medication called cyclosporine which is very effective at treating it.
Until you can get him to the vet for a test, I would buy some over the counter "artificial tears" and an eye drop called "celluvisc" to help keep the eye moist. The celluvisc lasts longer. You can get both at any pharmacy over the counter. Put it in his eye every 3-4 hours.
He could also have a viral infection in that eye.
But his breed is predisposed to dry eye with his large eyeballs and small eyelids.
www.VeterinaryPartner.com/Content.plx?P=A&A=1326 more on dry eye with pictures