Garment initially covering primate (4)
I believe the answer is:
cape
'garment' is the definition.
(cape is a kind of garment)
'initially covering primate' is the wordplay.
'initially' says to take the initial letters.
'primate' becomes 'ape' (large primate).
The first letter of 'covering' is 'c'.
'c'+'ape'='CAPE'
(Other definitions for cape that I've seen before include "Wrap" , "Outer garment" , "Head; garment" , "Cod possibly" , "Cloak or headland (4)" .)