Running behind time, heading off with great excitement (8)
I believe the answer is:
elatedly
'with great excitement' is the definition.
'running behind time heading off' is the wordplay.
'running behind time' becomes 'belatedly'.
'heading off' means to remove the first letter.
'belatedly' with its initial letter taken away is 'ELATEDLY'.
(Other definitions for elatedly that I've seen before include "showing great emotion" , "high spirits" , "Happily" , "Joyfully" .)