Go off with someone’s pet, perhaps, for a doze? (6)
I believe the answer is:
catnap
'for a doze?' is the definition.
(I have seen 'Brief doze' mean 'catnap' so perhaps 'doze' could also mean 'catnap')
I can't explain the remainder of the clue.
(Other definitions for catnap that I've seen before include "Forty winks (for moggy?)" , "(Tom's) snooze?" , "Short sleep" , "go out briefly" , "Kip" .)