Very little or slightly drunk (6)
I believe the answer is:
tiddly
This clue is a double definition.
'very little' is the first definition.
(tiddly means very small)
'slightly drunk' is the second definition.
(I've seen this before)
'or' acts as a link.
(Other definitions for tiddly that I've seen before include "Diminutive - drunk" , "Tiny" , "Merry" , "A little drunk" , "Slightly drunk (colloq.)" .)