Creep — boring thing following councillor (5)
I believe the answer is:
crawl
'creep' is the definition.
(I know that creep can be written as crawl)
'boring thing following councillor' is the wordplay.
'boring thing' becomes 'awl' (tool for boring holes).
'following' says to put letters next to each other.
'councillor' becomes 'cr' (abbreviation).
'awl' after 'cr' is 'CRAWL'.
(Other definitions for crawl that I've seen before include "Unusually slow rate of movement" , "Get along swimmingly" , "Baby-like advance" , "Way of swimming" , "but very slowly on land" .)