Launch unfinished biography on posh chap (4-3)
I believe the answer is:
lift-off
'launch' is the definition.
(thesaurus)
'unfinished biography on posh chap' is the subsidiary indication.
'unfinished' means to remove the last letter.
'biography' becomes 'life' ('life' can be a synonym of 'biography').
'on' is a charade indicator (letters next to each other).
'posh chap' becomes 'toff' (I've seen this before).
'life' with its last letter taken away is 'lif'.
'lif'+'toff'='liftoff'
(Other definitions for lift-off that I've seen before include "Start of rocket's journey" , "Launch (of rocket)" , "Moment of rocket's launch" , "Instant when rocket leaves launching pad" .)