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 don't know anything 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' is a charade indicator (letters next to each other).
'a cup' becomes 'cur' (I am not sure about this - if you are sure you should believe this answer much more).
'sit' written backwards gives 'tis'.
'tis' put after 'cur' is 'CURTIS'.
'to' is the link.
Can you help me to learn more?
(Another definition for curtis that I've seen is " Hollywood actor".)