Advance for a little bit (7)
I believe the answer is:
promote
'advance' is the definition.
(I know that advance can be written as promote)
'for a little bit' is the wordplay.
'for' becomes 'pro' (as in being 'pro' something).
'a little bit' becomes 'mote' (I've seen this in another clue).
'pro'+'mote'='PROMOTE'
(Other definitions for promote that I've seen before include "Publicise; raise in rank" , "Advocate, advance" , "Raise to higher rank" , "Support, encourage" , "Raise to a more senior position" .)