A bishop grabbing painter’s horse (4)
I believe the answer is:
arab
'horse' is the definition.
(Arab is a type of horse)
'a bishop grabbing painter's' is the wordplay.
'bishop' becomes 'B' (chess abbreviation).
'grabbing' indicates putting letters inside.
'painter' becomes 'RA' (member of the Royal Academy of Arts).
'a'+'b'='ab'
'ab' enclosing 'ra' is 'ARAB'.
(Other definitions for arab that I've seen before include "Native of North Africa" , "One from the Middle East, say" , "Person from the Middle East, say" , "Mount" , "Qatari or Omani, say" .)