The most common and most likely answer to this crossword clue is the 9 plus 6 letter word PROBOSCIS MONKEY
I've seen this crossword clue in Independent
P
R
O
B
O
S
C
I
S
M
O
N
K
E
Y
More clues leading to the result PROBOSCIS MONKEY
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 = proboscis monkey.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.