What is the medical term meaning origin of or producing pus?

I think the answer you want is pyogenesis, purulent, or suppuration. The Word parts pyo- and -pur- can both mean pus.

A rather disgusting mnemonic for remembering this a zombie has come home to his mother because she has baked him his favorite. Pure (-pur-) pus pie. Oh (pyo-) boy, yummy.