Primate runs after large bird (5)
I believe the answer is:
lemur
'primate' is the definition.
(lemur is a kind of primate)
'runs after large bird' is the wordplay.
'runs' becomes 'r' (cricket abbreviation).
'after' says to put letters next to each other.
'large' becomes 'L' (eg in clothes sizes).
'bird' becomes 'emu' (emu is a type of bird).
'l'+'emu'='lemu'
'r' put after 'lemu' is 'LEMUR'.
(Other definitions for lemur that I've seen before include "Tree-dwelling primate" , "Arboreal primate native to Madagascar" , "Social climber" , "Malagasy primate" , "Long-tailed primate" .)