Work quietly applied to hospital department for one's rival (8)
I believe the answer is:
opponent
'one's rival' is the definition.
'opponent' can be an answer for 'rival' (a rival is an opponent or enemy). I'm not sure about the 'one's' bit.
'work quietly applied to hospital department' is the wordplay.
'work' becomes 'op' (abbreviation for opus).
'quietly' becomes 'p' (musical symbol).
'applied' becomes 'on'.
'to hospital department' becomes 'ENT' (Ear, Nose and Throat department).
'op'+'p'+'on'+'ent'='OPPONENT'
'for' is the link.
Can you help me to learn more?
(Other definitions for opponent that I've seen before include "Adversary, competitor" , "member of other team" , "Contestant you're matched against" , "One competes against you" , "'Adversary, antagonist (8)'" .)