Passing on a tip which has worked for lots of kids I know. Sorry for unscientific language but it was recommended by my brother's doctor for their tiny baby so it is safe and works.
If a kid gets gummy eyes, it can sometimes be because their tear ducts aren't draining properly (often because their nose is bunged up with a cold so there isn't much drainage going on in general). Instead of leaping for antibiotic eye drops, you can try gently massaging up the sides of their nose towards their eyes (carefully, of course). This will often squeeze some gunk out of their tear ducts into their eyes, where it will just be discharged with other gunk and can be wiped away with cotton wool dipped in cooled boiled water. It can be a bit grim looking but better out than in, I say. Then their eyes can start to drain more naturally and get rid of any lingering nasties the way nature intended.
Hope it works for someone!