Crossword Genius

Stay after drink - that has a point (8,5)

Ross

I believe the answer is:

cocktail stick

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'a point' is the definition.
Both the answer and definition are singular nouns.
Maybe there's a link between them I don't understand?

'stay after drink that has' is the wordplay.
'stay' becomes 'stick' (synonyms).
'after drink' becomes 'cocktail' (cocktail is a kind of drink. I am not sure about the 'after' bit.).
'that has' says to put letters next to each other.
'stick' put after 'cocktail' is 'COCKTAIL STICK'.

Can you help me to learn more?

(Other definitions for cocktail stick that I've seen before include "one passing through bit of Leicester, perhaps?" , "narrow segment of wood" , "one going through Cheshire?" , "Aid for eating small party eats" , "Drinks party item" .)

I've seen this clue in The Daily Express.
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