Finished next, penning a note to increase demands (2,3,4)
I believe the answer is:
up the ante
'to increase demands' is the definition.
(I've seen this before)
'finished next penning a note' is the wordplay.
'finished' becomes 'up' (eg 'time's up' means the time is finished).
'next' becomes 'then' (both can mean subsequently).
'penning' means one lot of letters goes inside another.
'note' becomes 'te' (musical note).
'then' placed around 'a' is 'thean'.
'up'+'thean'+'te'='UP THE ANTE'
(Other definitions for up the ante that I've seen before include "increase risk" , "Increase what is at stake" , "Pet tuna he used to raise poker stakes" , "To increase demands" , "increase risks" .)