Godfather chopping tail off fowl (4)
I believe the answer is:
capo
'godfather' is the definition.
(both are types of mafia boss)
'chopping tail off fowl' is the wordplay.
'chopping tail off' means to remove the last letter.
'fowl' becomes 'capon' (capon is a kind of fowl).
'capon' with its final letter removed is 'CAPO'.
(Other definitions for capo that I've seen before include "Mafia boss" , "Mafia chief?" , "Mafia leader" , "crime boss" , "Leader of gang" .)