Mobster needs beer having eaten chicken (2,6)
I believe the answer is:
al capone
'mobster' is the definition.
'beer having eaten chicken' is the wordplay.
'beer' becomes 'ale' (I've seen this before).
'having eaten' indicates putting letters inside.
'chicken' becomes 'capon' (capon is a kind of chicken).
'ale' going around 'capon' is 'AL CAPONE'.
'needs' acts as a link.
(Other definitions for al capone that I've seen before include "Famous gangster" , "Chicago mobster jailed for tax evasion" , "Notorious Chicago gangster called Scarface" , "Hood" , "Notorious gangster, d. 1947" .)