Your kitten has a condition called conjunctivitis. Your veterinarian will be able to tell if the infection is bacterial or viral in nature, and that will determine how severe it is. Fortunately, many diseases can be treated at home with antibiotics and topical ointments such as Karo syrup (or corn syrup), which you administer by syringe. Please check out our article on treating conjunctivitis for more information on this topic.
Why does my cat’s eye bulge?
Many of your cats eyes may look slightly puffy after he wakes up from sleep due to extra fluid around his eyeball caused by inflammation there. This condition is called uveal glaucoma and usually doesn’t cause any problems so long as your pet isn’t sensitive to light or blinking too much; but if it seems like the swelling gets worse over time or his vision starts getting blurry then talk to your vet right away about having him evaluated for glaucoma treatment options such as medication drops into each eye once per week until they’re gone (and some even require surgery). If we had one piece of advice we’d give new cat parents—get those kittens checked early!