Personal assistant in street quarrel (4)
I believe the answer is:
spat
'quarrel' is the definition.
(I know that spat is a type of quarrel)
'personal assistant in street' is the wordplay.
'personal assistant' becomes 'PA' (abbreviation).
'in' is an insertion indicator.
'street' becomes 'st' (abbreviation).
'pa' placed within 'st' is 'SPAT'.
(Other definitions for spat that I've seen before include "Quarrel; oyster spawn" , "A small tiff or quarrel" , "Slight quarrel - ejected saliva" , "Pat's involved in petty quarrel" , "Taps in minor quarrel" .)