Old Scot going round looking embarrassed for project (7)
I believe the answer is:
predict
'for project' is the definition.
(thesaurus)
'old scot going round looking embarrassed' is the wordplay.
'old scot' becomes 'Pict' (Picts were an ancient people of Scotland).
'going round' indicates putting letters inside (some letters go around others).
'looking embarrassed' becomes 'red' (someone may go red when embarrassed).
'pict' going around 'red' is 'PREDICT'.
(Other definitions for predict that I've seen before include "Guess (the future)" , "Foresee" , "Prophesy - portend" , "Divine" , "Foretell, prophesy" .)