Dog belonging to rich owner (4)
I believe the answer is:
chow
'dog' is the definition.
(short for chow chow)
'belonging to rich owner' is the wordplay.
'belonging to' says the answer is hidden in the clue (I've seen this in other clues).
'CHOW' can be found hidden inside 'rich owner'.
(Other definitions for chow that I've seen before include "breakfast?" , "Mixed < dog" , "[HOUND]" , "Nosh" , "Long-haired dog of Chinese breed, or informal food" .)