Rush with the rent (4)
I believe the answer is:
tear
I believe this clue is a double definition.
'rush' is the first definition.
(tear is a kind of rush)
'the rent' is the second definition.
('tear' can be a synonym of 'rent')
'with' is the link.
(Other definitions for tear that I've seen before include "A rip and a sign of sadness" , "Lacerate - sign of sorrow" , "A rip goes on a binge?" , "Drop secreted by the lacrimal glands" , "Rend; travel in haste" .)