Primate's pet project suppressed by President (8)
I believe the answer is:
bushbaby
'primate's' is the definition.
(bushbaby is a kind of primate)
'pet project suppressed by president' is the wordplay.
'pet project' becomes 'baby' (I've seen this before).
'suppressed by' means one lot of letters go next to another.
'president' becomes 'bush' (George H W Bush or George W Bush).
'baby' after 'bush' is 'BUSHBABY'.
(Other definitions for bushbaby that I've seen before include "Small African lemur, like shrubbery infant" , "Small African nocturnal primate with large eyes" , "Nocturnal African lemur (closely related to one of two US presidents?)" , "Small, nocturnal, tree-dwelling primate of Africa with large eyes" , "African lemur-like primate (closely related to one of two US presidents?)" .)