I headed back to arrive at food store (4)
I believe the answer is:
deli
'food store' is the definition.
(short for delicatessen)
'i headed back' is the wordplay.
'headed' becomes 'led' ('lead' can be a synonym of 'head').
'back' is a reversal indicator.
'i'+'led'='iled'
'iled' backwards is 'DELI'.
'to arrive at' is the link.
(Other definitions for deli that I've seen before include "Cold-meat counter" , "Fine foods shop" , "Food store" , "Unusual food shop" , "place to buy food" .)