From what you wrote, it sounds possibly like an eye infection. There are drops you can buy for this in pet stores (called something like Clear Eyes). Chlorinated water will also irritate the eyes, but the white (pus?) coming from them sounds more like an eye infection. Does he have any other symptoms (open-mouth breathing, wheezing, white/mucus coming from the nose or mouth, tilting to one side in the water?
What have you been feeding him? I'd get him a submersible water heater to keep the temp constant at around 78F. Are you using chlorinated water where you are now? Your turtle needs a stable habitat and it sounds like he's being moved around too much for his own good.
BTW, I deleted your other post. Please do not cross post.
"You become responsible, forever, for what you have tamed." -Antoine de Saint Exupery-