Badger exterminator gutlessly follows nuisance (6)
I believe the answer is:
pester
'badger' is the definition.
('pester' can be a synonym of 'badger')
'exterminator gutlessly follows nuisance' is the wordplay.
'gutlessly' means to remove the middle letters.
'follows' is a charade indicator (letters next to each other).
'nuisance' becomes 'pest' (I've seen this before).
'exterminator' with its middle taken out is 'er'.
'er' put after 'pest' is 'PESTER'.
(Other definitions for pester that I've seen before include "Channel island" , "Harass" , "Trouble persistently" , "Trouble, annoy" , "Annoy with frequent requests" .)