Youth entertained in private engagement (4)
I believe the answer is:
teen
'youth' is the definition.
(teenager)
'entertained in private engagement' is the wordplay.
'entertained in' indicates a hidden word.
'TEEN' can be found hidden inside 'private engagement'.
(Other definitions for teen that I've seen before include "Juvenile" , "Youngster over twelve" , "Young person" , "Youth" , "Adolescent (informal)" .)