Touchy having failed to finish exam (4)
I believe the answer is:
test
'exam' is the definition.
('test' can be a synonym of 'exam')
'touchy having failed to finish' is the wordplay.
'touchy' becomes 'testy'.
'having failed to finish' means to remove the last letter.
'testy' with its final letter taken away is 'TEST'.
(Other definitions for test that I've seen before include "Trial; cricket match" , "Short exam" , "Experiment" , "Check" , "Dip one's toe in" .)