Right on to succeeding first prize, and quickly! (6)
I believe the answer is:
pronto
'quickly' is the definition.
(I've seen this in another clue)
'right on to succeeding first prize' is the wordplay.
'right' becomes 'r' (common abbreviation).
'succeeding' means one lot of letters go next to another (succeeding can mean going after or following).
'first prize' becomes 'p' (1st letter of 'prize').
'r'+'on'+'to'='ronto'
'ronto' after 'p' is 'PRONTO'.
'and' is the link.
(Other definitions for pronto that I've seen before include "Quickly, immediately" , "Double quick!" , "Right now, immediately" , "Quick as you can, amigo" , "Italian telephonic greeting" .)