Succeeded in project by a whisker (4)
I believe the answer is:
just
'by a whisker' is the definition.
(I've seen this before)
'succeeded in project' is the wordplay.
'succeeded' becomes 's' (genealogical abbreviation).
'in' is an insertion indicator.
'project' becomes 'jut' (I've seen this before).
's' inserted into 'jut' is 'JUST'.
(Other definitions for just that I've seen before include "And nothing more" , "Merited" , "Barely - equitable" , "Fair-minded" , "Recently" .)