Stops for a cigarette puff and a drink (5,2)

'stops' is the definition.
(drawing up is a kind of stopping)

'cigarette puff and a drink' is the wordplay.
I cannot quite understand how this works, but
'puff' could be 'draw' (synonyms) and 'draw' is found within the answer.
'drink' could be 'sup' (to sup is to drink) and 'sup' is present in the answer.
No letters remain.
This may be the basis of the clue (or it may be nonsense).

'for a' acts as a link.

Can you help me to learn more?