Brought up on food, say (4)
I believe the answer is:
bred
'brought up' is the definition.
(to breed is to bring up or raise)
'food say' is the wordplay.
'food' becomes 'bread' (bread is a type of food).
'say' shows a homophone (sound like).
'bread' sounds like 'BRED'.
'on' is the link.
(Other definitions for bred that I've seen before include "Produced offspring" , "Bore" , "Brought forth" , "Developed (animal variety)" , "Generated" .)