The fight by a mug to take credit (5)
I believe the answer is:
scrap
'the fight' is the definition.
(a scrap is a minor fight)
'a mug to take credit' is the wordplay.
'a mug' becomes 'sap'.
'to take' is an insertion indicator (inserted letters are taken in).
'credit' becomes 'cr' (abbreviation).
'sap' enclosing 'cr' is 'SCRAP'.
'by' acts as a link.
(Other definitions for scrap that I've seen before include "Waste metal or slang fight" , "boxing match?" , "Fragment or remnant" , "Discard the small fragment" , "Oddment" .)