Not quietly, removes skin from fish (4)
I believe the answer is:
eels
'fish' is the definition.
(eel is a type of fish)
'not quietly removes skin' is the wordplay.
'not' indicates named letters should be taken away.
'quietly' becomes 'p' (p for piano and quiet).
'removes skin' becomes 'peels' (I've seen this before).
'peels' with 'p' removed is 'EELS'.
'from' is the link.
(Other definitions for eels that I've seen before include "Slippery creatures" , "Fish (that may be jellied)" , "Water snakes twisting in lees" , "Slithery fish" , "Jellied --, traditional dish" .)