Provide more of drink, holding favourite item back (4,2)
I believe the answer is:
step up
'provide more' is the definition.
(both can mean to increase something)
'drink holding favourite item back' is the wordplay.
'drink' becomes 'sup' (I've seen this before).
'holding' is an insertion indicator.
'favourite item' becomes 'pet'.
'back' is a reversal indicator.
'pet' written backwards gives 'tep'.
'sup' enclosing 'tep' is 'STEP-UP'.
'of' acts as a link.
(Other definitions for step up that I've seen before include "Come forward" , "Increase as with pace - or ascend stair?" , "Approach" , "- Bezukhov (War and Peace)" , "Accelerate - promotion" .)