Unpleasant characters regularly seen in Burghley (4)
I believe the answer is:
ugly
'unpleasant' is the definition.
(I've seen this in another clue)
'characters regularly seen in burghley' is the wordplay.
'characters regularly seen in' means one should take alternating letters (letters selected at regular intervals).
The alternate letters of 'burghley' are 'UGLY'.
(Other definitions for ugly that I've seen before include "Hideous" , "Deficient in beauty" , "Nasty to look at" , "Visually unpleasant; dangerous (situation)" , "Repulsive to the eye" .)