Finishes milk drink (4,2)
I believe the answer is:
uses up
'finishes' is the definition.
'milk drink' is the wordplay.
'milk' becomes 'use' (I've seen this in other clues).
'drink' becomes 'sup' (I've seen this before).
'use'+'sup'='USES UP'
(Other definitions for uses up that I've seen before include "Consumes" , "'Depletes (4,2)'" , "Exhausts" , "drains" .)