Conjunctivitis is quite common in kittens and is treated with prescribed antibiotics. It is contagious (spread by contact) and can be transmitted from kitten to human and vice versa. Good hygiene is essential to prevent cross-infection. Wash your hands/any utensils after handling or feeding an infected kitten. Gummy eyes are also associated with Cat Flu, more common in stray or feral kittens as the mothers are unlikely to have been vaccinated.
Note: Conjunctivitis in humans needs prompt treatment as it can become very painful. On the plus side, after many years of handling gummy-eyed kittens I have thankfully become almost immune to many of the bacteria involved (i.e. my own immune system deals with them swiftly).
A few kittens’ eyes open prematurely and there is a risk that they will dry out so regular eye drops are needed. Many such kittens do not survive. Sometimes, a vet will stitch the lid closed temporarily. Entropion (turning in of the eyelids so that lashes scrape the cornea) also leads to discharge.
Ultra-typed short-nosed or flat-faced breeds (Persians, Exotics) may be prone to blocked tear ducts and some lack tear ducts entirely.
If the eyes fail to open, check for micropthalmia (under-developed eyeballs) or anopthalmia (no eyeballs at all). Kittens with micropthalmia may have some vision. Whether or not to rear blind or partially sighted kittens depends on the likelihood of them finding permanent homes. Blind cats can lead happy lives if the household is adapted to their needs.