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' says to put letters next to each other.
'president' becomes 'bush' (George H W Bush or George W Bush).
'baby' put 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?)" .)