Note non-drinker is concealing their hand, perhaps (4)

'hand perhaps' is the definition.
'mitt' can be an answer for 'hand'. I am unsure of the 'perhaps' bit.

'note non-drinker is concealing' is the wordplay.
I cannot quite see how this works, but
'note' could be 'mi' (as in do-re-mi) and 'mi' is found within the answer.
'non-drinker' could be 'tt' (abbreviation for teetotaller) and 'tt' is located in the answer.
This accounts for all the letters.
This may be the basis of the clue (or it may be nonsense).

'their' is the link.

Can you help me to learn more?