Digger by street causing bother (6)
I believe the answer is:
molest
'bother' is the definition.
(I know that molest is a more specific form of the action bother)
'digger by street' is the wordplay.
'digger' becomes 'mole' (I've seen this before).
'by' means one lot of letters go next to another.
'street' becomes 'st' (abbreviation).
'mole'+'st'='MOLEST'
'causing' is the link.
(Other definitions for molest that I've seen before include "Maliciously interfere with" , "Bug" , "Plague" , "Harass" , "Attack sexually" .)