I have used Antibiotic Ointment lots of times for my cats and kittens gunky eyes. Make sure you get the plain stuff, non with the pain reliever just the basic. I have even put in in my eye. When ever I had a cat with an eye that looked like it was irritated I would put a dab on the outside corner of the eye so he could blink it in. It has never bothered them and has always worked wonders. I usually do it for three days in a row and then they are good to go then.
Right now I have a feral cat in my a back yard with a gunky looking eye. It started out as a clear yellow discharge with red sometimes. He then started keeping it closed all the time. It has gotten really bad looking and seems to be moving into his other eye. I can't touch him he's too skiddish. One day I was able to prepare myself with a dab of ointment on my finger and I got it on him. His eye looked way better the next day but since then I am losing the catch a cat battle. I ordered Fishmox and it's on its way. Otherwise he may keep suffering and go blind.
pet eye problems
a kitten that I rescued 2 or 3 years ago( I still have him) kept having eye problems. For 2 or 3 weeks I would clean his eyes with a warm wet wash cloth. They were better but not cured. Someone told me to put antibiotic ointment just under the eyes being careful not to get it in the eyes. Just put some on your finger and rub it in the fur around the bottom of the eye. I did this 3 times a day for about a week or so and his eyes were cured! No more problems.