Shop one managed to put up (4)
I believe the answer is:
deli
'shop' is the definition.
(deli is a kind of shop)
'one managed to put up' is the wordplay.
'one' becomes 'I' (Roman numeral).
'managed' becomes 'led'.
'to put up' says the letters should be written backwards.
'i'+'led'='iled'
'iled' backwards is 'DELI'.
(Other definitions for deli that I've seen before include "Specialised food shop" , "Specialist food shop" , "Food shop (colloq.)" , "Food store" , "Cold-meat counter" .)