Slight quarrel with father in the street (4)
I believe the answer is:
spat
'slight quarrel' is the definition.
(I've seen this before)
'father in the street' is the wordplay.
'father' becomes 'pa' (pa can mean dad or father).
'in the' is an insertion indicator.
'street' becomes 'st' (abbreviation).
'pa' inserted into 'st' is 'SPAT'.
'with' acts as a link.
(Other definitions for spat that I've seen before include "cross words" , "Taps in a small quarrel" , "Little quarrel - leg wear" , "Expectorated in slight quarrel" , "A minor dispute or row" .)