Lass finishes off black tea: not nice (4)
I believe the answer is:
kate
'lass' is the definition.
(woman's name)
'finishes off black tea not nice' is the wordplay.
'finishes off' indicates one should take the final letters (take the finishing letters off of the words).
The last letters of 'black tea not nice' are 'KATE'.
(Other definitions for kate that I've seen before include "'Kiss me, ---' (Shrew)" , "'Kiss me ___' (show)" , "Cole Porter musical 'Kiss Me ....'" , "Katherine (dim.)" .)