What make your eyes run something that looks like pus?

Eye discharge. You may have conjunctivitis. If the discharge is very milky, then seek help from your local ophthalmologist. Bacterial conjunctivitis will require a topical antibiotic eyedrop to help clear the infection. Benign causes of eye discharge include dry eye and allergies.
Pus like discharge. Yellow green copious pus like discharge is likely bacterial conjunctivitis and treated w fluoroquinolone antibiotic 4x a day.