Leaving? Rousing cheer initially let out (7)
I believe the answer is:
exiting
'leaving?' is the definition.
('exit' can be a synonym of 'leave')
'rousing cheer initially let out' is the wordplay.
I cannot quite understand how this works, but
'out' could be 'exit' (I've seen this in another clue) and 'exit' is found within the answer.
This may be the basis of the clue (or it may be nonsense).
(Other definitions for exiting that I've seen before include "Leaving" , "Going out" .)