Sometime the morning after, having the test (4)
I believe the answer is:
exam
'the test' is the definition.
('exam' can be a synonym of 'test')
'sometime the morning after' is the wordplay.
'sometime' becomes 'ex' (prefix meaning former or onetime).
'the morning' becomes 'am'.
'after' is a charade indicator (letters next to each other).
'ex'+'am'='EXAM'
'having' is the link.
(Other definitions for exam that I've seen before include "(Academic) test" , "In brief, a test as in school" , "Test (abbr)" , "Educational test" , "Important test" .)