Stay with lute playing in a shrewd manner (8)
I believe the answer is:
astutely
'in a shrewd manner' is the definition.
(I've seen this before)
'stay with lute playing' is the wordplay.
'with' is a charade indicator (letters next to each other).
'playing' indicates anagramming the letters.
'stay'+'lute'='staylute'
'staylute' with letters rearranged gives 'ASTUTELY'.
(Other definitions for astutely that I've seen before include "Shrewdly" , "In a shrewd manner" , "cannily" .)