Bit of bacon some won't eat — rejected if not Danish, primarily (4)
I believe the answer is:
rind
'bit of bacon some won't eat' is the definition.
(some people cut the rind off bacon)
'rejected if not danish primarily' is the wordplay.
'primarily' suggests taking the first letters.
The first letters of 'rejected if not danish' is 'RIND'.
(Other definitions for rind that I've seen before include "tough skin" , "Fruit skijn as of orange" , "Outer skin of citrus fruit" , "Tough outer layer" , "Zest" .)