Player with puffy cheeks (5)
I believe the answer is:
'player' is the definition.
(piper is a kind of player)
I cannot understand how the rest of the clue works.
Can you help me to learn more?
(Other definitions for piper that I've seen before include "Military musician" , "Highland instrumentalist" , "'Liam O'Flynn or Seamus Ennis, say (5)'" , "Musician (paid for a tune?)" , "Wind instrumentalist" .)