A slight cut (4)
I believe the answer is:
snub
I believe this clue is a double definition.
'a slight' is the first definition.
(both can mean to treat someone disrespectfully)
'cut' is the second definition.
(I know that cut can be written as snub)
(Other definitions for snub that I've seen before include "'Rebuff, ignore (4)'" , "Rebuff humiliatingly" , "Rude put-down; blunt (nose)" , "Reject a person with cold behaviour" , "Shun" .)