Drink finished, monarch leaves for the city (5)
I believe the answer is:
kirov
'the city' is the definition.
(I have seen 'Russian city' mean 'kirov' so perhaps 'city' could also mean 'kirov')
'drink finished monarch leaves' is the wordplay.
'drink' becomes 'kir' (type of cocktail).
'finished' becomes 'over' (similar in meaning).
'monarch' becomes 'ER' (abbreviation for Elizabeth Regina, British Queen).
'leaves' is a deletion indicator.
'over' with 'er' removed is 'ov'.
'kir'+'ov'='KIROV'
'for' is the link.
(Other definitions for kirov that I've seen before include "Stalinist" , "Soviet city" , "Russian city and ballet company" , "Famous ballet" , "Russian dancers" .)