Polly has constantly runny gungy eyes, she's had them since I took her in back in '93. My vet, who has a speciality in opthalmics, has checked them and thinks that her tear ducts have become blocked due to cat flu (before I became her slave!). He tested for chlamydia, but she was negative.
I wipe her eyes with clean tissue, which she loves, as I stroke her cheeks and nose at the same time. She's 21, so this hasn't shortened her life or caused any noticeable problems with her eyesight - vet checked this also.
My advice is: not to worry, just wipe her eyes with tissues when they look mucky. Polly has got into the habit of coming to me when she wants her eyes wiped, maybe Jonty will too.