Hard losing initial argument (4)
I believe the answer is:
tiff
'argument' is the definition.
(I've seen this before)
'hard losing initial' is the wordplay.
'hard' becomes 'stiff' (similar in meaning).
'losing initial' means to remove the first letter.
'stiff' with its initial letter taken away is 'TIFF'.
(Other definitions for tiff that I've seen before include "Petty squabble" , "A spat or small row" , "Hard to lose opening argument" , "causing quarrel" , "Mild quarrel" .)