No pet dog having to sit up with a cup (6)
I believe the answer is:
curtis
'no pet dog having' is the definition.
I know nothing about this answer so I can't tell whether this works.
'sit up with a cup' is the wordplay.
'up' is a reversal indicator (in down clue: letters go upwards).
'with' says to put letters next to each other.
'a cup' becomes 'cur' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'sit' written backwards gives 'tis'.
'tis' put after 'cur' is 'CURTIS'.
'to' acts as a link.
Can you help me to learn more?
(Another definition for curtis that I've seen is " Hollywood actor".)