Partly cover finished drink (7)
I believe the answer is:
overlap
'partly cover' is the definition.
(I've seen this before)
'finished drink' is the wordplay.
'finished' becomes 'over' ('over' can be similar in meaning to 'finished').
'drink' becomes 'lap' (lapping is a kind of drinking**).
'over'+'lap'='OVERLAP'
(Other definitions for overlap that I've seen before include "Common area of interest" , "Partially cover" , "Outflank" , "Partly hide" , "Extend on top of" .)