Detective given largely small piece of food (5)
I believe the answer is:
morse
'detective' is the definition.
(Inspector Morse, fictional detective)
'largely small piece of food' is the wordplay.
'largely' means to remove the last letter (most but not all of the letters).
'small piece of food' becomes 'morsel' (I've seen this before).
'morsel' with its last letter removed is 'MORSE'.
'given' is the link.
(Other definitions for morse that I've seen before include "TV Oxford cop" , "Fictional inspector; code" , "Radio code inventor" , "Oxford detective" , "10 [SLEUTH]" .)