Note written by one's European shrink (8)
I believe the answer is:
minimise
'shrink' is the definition.
(to minimise is to reduce or shrink)
'note written by one's european' is the wordplay.
'note' becomes 'minim' (type of musical note).
'written by' means one lot of letters go next to another.
'one's' becomes 'I's' (Roman numeral).
'european' becomes 'E' (abbreviation e.g. EU).
'minim'+'is'+'e'='MINIMISE'
(Other definitions for minimise that I've seen before include "Scale down" , "Semi I'm in (anag)" , "Diminish" , "Downplay, understate" , "Shrink" .)