Half-heartedly holding back cool old city (10)
I believe the answer is:
stalingrad
'old city' is the definition.
(former name of Volgograd in Russia)
'half-heartedly holding back cool' is the wordplay.
'half-heartedly holding back' becomes 'staling' (I am not sure about this - if you are sure you should believe this answer much more).
'cool' becomes 'rad' (I've seen this in another clue).
'staling'+'rad'='STALINGRAD'
(Other definitions for stalingrad that I've seen before include "where food was short during siege" , "Russian scene of major World War two battle" , "Former name of Volgograd" , "Old Russian city" .)