Rabbit's tail shortened near beginning of surgery (4)
I believe the answer is:
scut
'rabbit's tail' is the definition.
(scut can mean a rabbit's tail)
'shortened near beginning of surgery' is the wordplay.
'shortened' becomes 'cut' (synonyms).
'near' says to put letters next to each other (I've seen this in other clues).
'beginning of' indicates taking the first letters.
The initial letter of 'surgery' is 's'.
'cut' put after 's' is 'SCUT'.
(Other definitions for scut that I've seen before include "Short tail of hare" , "(Hare's) short upright tail" , "Reindeer tail" , "Reindeer's tail" , "Small tail as of rabbit" .)