Shot out of the job-centre in the city (4)
I believe the answer is:
troy
'city' is the definition.
(Troy is an example)
'shot out of the job-centre' is the wordplay.
'shot' becomes 'try' (I've seen this before**).
'out of' indicates putting letters inside.
'centre' means to look at the middle letters.
The middle letter of 'job' is 'o'.
'try' enclosing 'o' is 'TROY'.
'in the' is the link.
(Other definitions for troy that I've seen before include "Helen of --, beauty" , "Wooden horse siege city" , "part of Asia Minor" , "Hardy sergeant" , "Ancient city where fair Helen was a captive" .)