Sweetheart goes off backing shop (5)
I believe the answer is:
store
'shop' is the definition.
(I know that shop can be written as store)
'sweetheart goes off backing' is the wordplay.
'sweetheart' becomes 'e' (the heart/centre of 'sweet').
'goes off' becomes 'rots' (go off can mean to rot or decay).
'backing' shows that the letters should be reversed in order.
'e'+'rots'='erots'
'erots' written backwards gives 'STORE'.
(Other definitions for store that I've seen before include "Large shop; keep" , "Large shop; computer memory" , "Reserve, shop" , "Shop; repository" , "Computer memory; supply kept for when needed" .)