Burden young offender with some medicine? Not at first (6)
I believe the answer is:
impose
'burden' is the definition.
(thesaurus)
'young offender with some medicine? not at first' is the wordplay.
'young offender' becomes 'imp' (I've seen this in other clues).
'with' means one lot of letters go next to another.
'some medicine?' becomes 'dose' (dose is a kind of medicine).
'not at first' means to remove the first letter.
'dose' with its initial letter taken away is 'ose'.
'imp'+'ose'='IMPOSE'
(Other definitions for impose that I've seen before include "Decree" , "Force on or take advantage of" , "Exact or levy, say tax" , "Force (on someone)" , "Inflict (something unpleasant)" .)