British backing also affected military training centre (4,4)
I believe the answer is:
boot camp
'military training centre' is the definition.
(camp for training army recruits)
'british backing also affected' is the wordplay.
'british' becomes 'b' (abbreviation e.g. in 'BBC').
'backing' shows that the letters should be reversed in order.
'also' becomes 'too' (synonyms).
'affected' becomes 'camp' (camp can mean exaggerated or affected).
'too' backwards is 'oot'.
'b'+'oot'+'camp'='BOOT CAMP'
(Other definitions for boot camp that I've seen before include "where terrier potentially goes" , "US training centre for military recruits" , "Harsh (US) military training establishment" , "Centre for juvenile offenders" , "Scene of strict discipline for young offenders" .)