Quietly finishing off drink, or what’s left? (4)
I believe the answer is:
rump
'what's left?' is the definition.
(rump can mean the remainder)
'quietly finishing off drink' is the wordplay.
'quietly' becomes 'p' (p for piano and quiet).
'finishing off' means one lot of letters go next to another (some letters finish or go after others).
'drink' becomes 'rum' (rum is a kind of drink).
'p' after 'rum' is 'RUMP'.
'or' is the link.
(Other definitions for rump that I've seen before include "Posterior" , "Fleshy hindquarters" , "Animal's hind quarters" , "Meat (from behind?)" , "Cut of meat from the rear" .)