Rush to run heats (5)
I believe the answer is:
haste
'rush' is the definition.
('haste' can be a synonym of 'rush')
'run heats' is the wordplay.
'run' indicates anagramming the letters (I've seen this in other clues).
'heats' is an anagram of 'HASTE'.
'to' is the link.
(Other definitions for haste that I've seen before include "The more of it, the less speed, it is said" , "Speed, hurry" , "''More . . . . ., less speed'' (5)" , "Leader abandons modest" , "Expeditiousness" .)