The dog has to sit up with a cup (6)
I believe the answer is:
curtis
'the' is the definition.
I don't know anything about this answer so I cannot judge whether this works.
'dog has to sit up with a cup' is the wordplay.
I cannot quite see how this works, but
'dog' could be 'cur' (both can mean a despicable person) and 'cur' is found in the answer.
an anagram of 'sit' is 'tis' which is within the answer.
No letters remain.
This explanation may well be incorrect...
Can you help me to learn more?
(Another definition for curtis that I've seen is " Hollywood actor".)