Certainly not on the sly (8)
I believe the answer is:
honestly
'certainly' is the definition.
Both the definition and answer are adverbs. Perhaps you can see a link between them that I can't see?
'not on the sly' is the wordplay.
'not' becomes 'hone' (I am not sure about this - if you are sure you should believe this answer much more).
'on' means one lot of letters goes inside another (as in clothing 'on' a person).
'the' becomes 't' (the is pronounced as a 't' sound in some dialects).
't' placed within 'sly' is 'stly'.
'hone'+'stly'='HONESTLY'
Can you help me to learn more?
(Other definitions for honestly that I've seen before include "In a just manner" , "In a truthful way" , "Frankly -- that's outrageous!" , "Fairly" , "On the sly (anag.)" .)