Dog eating primarily minced meat (4)
I believe the answer is:
lamb
'meat' is the definition.
(lamb is a kind of meat)
'dog eating primarily minced' is the wordplay.
'dog' becomes 'lab' (short for labrador).
'eating' is an insertion indicator.
'primarily' says to take the initial letters.
The first letter of 'minced' is 'm'.
'lab' going around 'm' is 'LAMB'.
(Other definitions for lamb that I've seen before include "Writer" , "A follower" , "Meaty" , "Young shee" , "Young sheep" .)