A little hasty exposing swelling (4)
I believe the answer is:
stye
'swelling' is the definition.
(stye is a kind of swelling)
'a little hasty exposing' is the wordplay.
'a little' indicates the answer is hidden within the clue.
'STYE' is hidden in the letters of 'hasty exposing'.
(Other definitions for stye that I've seen before include "Inflammation on eyelid" , "Red eye" , "Infected swelling of the eye" , "Climb" , "Eye complaint" .)