Cheeky young lad involved in raising cheeky theory (4)
I believe the answer is:
tyke
'cheeky young lad' is the definition.
(tyke can mean a mischievous child)
'involved in raising cheeky theory' is the wordplay.
'raising' is a reversal indicator (in a down clue, letters go up).
'involved in' indicates the answer is hidden within the clue.
'ekyt' is hidden in the letters of 'cheeky theory'.
'ekyt' backwards is 'TYKE'.
(Other definitions for tyke that I've seen before include "Colloquially, a small naughty child" , "Rough-mannered person - cheeky child" , "Cheeky child or small dog" , "Uncouth child" , "23 [BRAT]" .)