The most common and most likely answer to this crossword clue is the 6 letter word BONOBO
B
O
N
O
B
O
More clues leading to the result BONOBO
Buddy explains!

The straight part
We know that "Primate" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case primate = bonobo.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does BONOBO mean?
BONOBO noun- The pygmy chimpanzee, Pan paniscus, from Africa south of the Congo river.