Antelope that’s primarily Kenyan, unknown down under (4)
I believe the answer is:
kudu
'antelope' is the definition.
(kudu is a kind of antelope)
'primarily kenyan unknown down under' is the wordplay.
'primarily' indicates taking the first letters.
The initial letters of 'kenyan unknown down under' is 'KUDU'.
'that's' acts as a link.
(Other definitions for kudu that I've seen before include "Antelope with spiral horns" , "Spiral-horned antelope" , "African antelope with long spiralling horns" .)