Drone over South Island, say, causes hassle persistently (7)
I believe the answer is:
besiege
'hassle persistently' is the definition.
'drone over south island say' is the wordplay.
'drone' becomes 'bee'.
'over' is an insertion indicator.
'south' becomes 's' ('S' can be a synonym of 'south').
'island' becomes 'i'.
'say' becomes 'eg'.
's'+'i'+'eg'='sieg'
'bee' enclosing 'sieg' is 'BESIEGE'.
'causes' is the link.
(Other definitions for besiege that I've seen before include "Pester" , "Surround (a place) with hostile intent" , "Surround - harass" , "Invest" , "Encircle as a military tactic" .)