Story about backing to have importance (7)
I believe the answer is:
account
'story' is the definition.
(synonyms)
'about backing to have importance' is the wordplay.
'about' becomes 'ca' (short for circa).
'backing' shows that the letters should be reversed in order.
'to have importance' becomes 'count' (count can mean to matter or be important).
'ca' written backwards gives 'ac'.
'ac'+'count'='ACCOUNT'
(Other definitions for account that I've seen before include "Bill - story" , "Record of an event" , "judge" , "News report" , "Profitable to turn to this" .)