Pilot bothered by drills (6)
I believe the answer is:
aviate
'pilot' is the definition.
(synonyms)
'bothered by drills' is the wordplay.
'bothered' becomes 'ate' (something worrying is said to eat at you).
'by' becomes 'via' (travelling by or via a place).
'drills' indicates putting letters inside (to drill can mean to make a hole in something).
'ate' going around 'via' is 'AVIATE'.
(Other definitions for aviate that I've seen before include "Handle jumbo, maybe" , "Fly" , "Pilot a plane" , "Ride the Zephyr, say" .)