Rushed right in, late back (6)
I believe the answer is:
darted
'rushed' is the definition.
(darting is a kind of rushing)
'right in late back' is the wordplay.
'right' becomes 'rt'.
'in' is an insertion indicator.
'late' becomes 'dead' ('dead' can be similar in meaning to 'late').
'back' says the letters should be written backwards.
'dead' back-to-front is 'daed'.
'rt' put within 'daed' is 'DARTED'.
(Other definitions for darted that I've seen before include "Moved quickly and suddenly" , "Sniped" , "Moved suddenly and rapidly" , "Dashed on the Dublin transport system?" , "Dashed, flitted" .)