There is a very common eye infection in Thailand called Red Eye in Thai (ตาแดง). It makes the infected eye very red, itchy and there's pus excreted. It's very contagious and I've had it myself years ago.

Would anyone know the English name for this illness?

Conjunctivitis.

There are two basic forms of conjunctivitis: bacterial and viral. If there's pus it's probably bacterial and can be treated with antibiotics. No pus, probably viral and just needs to be left to run its course.