Priest left with a degree (4)
I believe the answer is:
lama
'priest' is the definition.
(lama is a kind of priest)
'left with a degree' is the wordplay.
'left' becomes 'l' (common abbreviation).
'with' is a charade indicator (letters next to each other).
'degree' becomes 'ma' (Master of Arts is a type of degree**).
'l'+'a'+'ma'='LAMA'
(Other definitions for lama that I've seen before include "Holy man in Tibet" , "Tibetan high priest" , "A Buddist monk in Tibet" , "devout Buddhist" , "A Buddhist priest in Tibet (4)" .)