Drip . . . or drill (4)
I believe the answer is:
bore
I believe this clue is a double definition.
'drip' is the first definition.
(I've seen this in another clue)
'drill' is the second definition.
(synonyms)
(Other definitions for bore that I've seen before include "Push through a crowd" , "Tiresome person - tidal flood" , "Drill or be tedious" , "Diameter of a gun barrel" , "Irritate" .)