Happy, having gone without one young man! (4)
I believe the answer is:
glad
'happy' is the definition.
('glad' can be a synonym of 'happy')
'gone without one young man' is the wordplay.
'without' indicates named letters should be taken away.
'young man' becomes 'lad' (I've seen this before).
'gone' with 'one' removed is 'g'.
'g'+'lad'='GLAD'
'having' acts as a link.
(Other definitions for glad that I've seen before include "Elated" , "More than willing" , "Feeling pleasure" , "Not sorry" , "Pleased, delighted" .)