Lady's half cut after quite strange drink (7)
I believe the answer is:
tequila
'drink' is the definition.
(tequila is a kind of drink)
'lady's half cut after quite strange' is the wordplay.
'half cut' suggests halving the letters.
'after' is a charade indicator (letters next to each other).
'strange' indicates anagramming the letters.
'lady' with half the letters taken is 'la'.
'quite' with letters rearranged gives 'tequi'.
'la' after 'tequi' is 'TEQUILA'.
(Other definitions for tequila that I've seen before include "Is there another drink to equal it?" , "Distilled agave drink" , "Mexican alcoholic drink" , "Equal it! (anag)" , "Cocktail -- 1988 American crime film -- equaliser units (anag)" .)