Unfinished pint and a soft drink -- beginning to desire a cocktail (4,6)
I believe the answer is:
pina colada
'cocktail' is the definition.
(I've seen this before)
'unfinished pint and a soft drink beginning to desire a' is the wordplay.
'unfinished' means to remove the last letter.
'and' says to put letters next to each other.
'soft drink' becomes 'cola' (cola is a kind of soft drink**).
'beginning to' says to take the initial letters.
The initial letter of 'desire' is 'd'.
'pint' with its final letter taken off is 'pin'.
'pin'+'a'+'cola'+'d'+'a'='PINA COLADA'
(Other definitions for pina colada that I've seen before include "Cocktail of rum, pineapple juice and coconut" , "rum-based potation" , "alcoholic one" , "Drink made of pineapple juice, coconut cream and rum" , "Perhaps contents of cocktail glass" .)