Runs home, disheartened with South American primate (6)
I believe the answer is:
rhesus
'primate' is the definition.
(rhesus is a kind of primate)
'runs home disheartened with south american' is the wordplay.
'runs' becomes 'r' (cricket abbreviation).
'disheartened' says to hollow out the word (remove centre letters).
'with' is a charade indicator (letters next to each other).
'south american' becomes 'sus' (I've seen this in other clues).
'home' with its middle removed is 'he'.
'r'+'he'+'sus'='RHESUS'
(Other definitions for rhesus that I've seen before include "Small macaque monkey" , "Blood type or kind of monkey (6)" , "Rushes (anag.)" , "Primate" , "Kind of monkey (or factor)" .)