Mule runs to right of primate (5)
I believe the answer is:
lemur
'primate' is the definition.
(I know that lemur is a type of primate)
'mule runs to right' is the wordplay.
'runs' is an anagram indicator.
'to right' becomes 'r' (common abbreviation).
'mule' with letters rearranged gives 'lemu'.
'lemu'+'r'='LEMUR'
'of' is the link.
(Other definitions for lemur that I've seen before include "Tree-living primate" , "Monkey-like animal of Madagascar" , "A primate; sounds like a S. Am. city" , "Arboreal mammal common to Madagascar" , "Madagascan primate" .)