I have a cat with a damaged eye (sometime before we adopted him) who frequently has discharge. Like Winic1, our vet recommended washing the eye several times a day with a warm wash cloth. He said to massage the eye as best you can to help open the duct and keep the eye clear to avoid infection. I gently hold Mac from the scruff of the neck...just enough to keep him still. Then I can wash/massage his eye with the wash cloth. I make the water pretty warm, but not too hot. He doesn't like at first but then doesn't put up a fuss after a minute or two. I think it might actually be soothing to him, since it's irritated.

The color of drainage can indicate infection. Anything yellow, green, or cloudy should be assessed by the vet right away.