Dad in street quarrel (4)
I believe the answer is:
spat
'quarrel' is the definition.
(spat is a kind of quarrel)
'dad in street' is the wordplay.
'dad' becomes 'pa' ('pa' can be a synonym of 'dad').
'in' indicates putting letters inside.
'street' becomes 'st'.
'pa' put inside 'st' is 'SPAT'.
(Other definitions for spat that I've seen before include "Minor argument" , "Small altercations between Pats" , "Trivial quarrel" , "Expelled" , "A small tiff or quarrel" .)