Rush at a mad rate (4)
I believe the answer is:
tear
'rush' is the definition.
(I know that tear is a more specific form of the action rush)
'a mad rate' is the wordplay.
'a mad' is an anagram indicator.
'rate' with letters rearranged gives 'TEAR'.
'at' acts as a link.
(Other definitions for tear that I've seen before include "Move quickly; pull apart" , "Droplet from the eye" , "Secretion of the lacrymal glands" , "speed" , "Rip, rend" .)