A dog has to sit up with this cup (6)
I believe the answer is:
curtis
'up with this cup' is the definition.
I don't know anything about this answer so I can't judge whether this works.
'a dog has to sit' is the wordplay.
I cannot really see how this works, but
'dog' could be 'cur' (a cur is an aggressive or scruffy dog) and 'cur' is found within the answer.
an anagram of 'sit' is 'tis' which is within the answer.
This accounts for all the letters.
This explanation may well be incorrect...
Can you help me to learn more?
(Another definition for curtis that I've seen is " Hollywood actor".)