Traveller with black dog (3-6)
I believe the answer is:
jet-setter
'traveller' is the definition.
(I have seen 'Wealthy traveller' mean 'jet setter' so perhaps 'traveller' could also mean 'jet setter')
'with black dog' is the wordplay.
'with' says to put letters next to each other.
'black' becomes 'jet' (similar in meaning).
'dog' becomes 'setter' (setter is a type of dog).
'jet' put next to 'setter' is 'JET-SETTER'.
(Other definitions for jet-setter that I've seen before include "Person who travels widely and frequently for pleasure" , "Member of the rich and famous who travel widely" , "Rich and fashionable traveller" .)