Note their fancy: hesitate (6)
I believe the answer is:
dither
'hesitate' is the definition.
(I've seen this before)
'note their fancy' is the wordplay.
'note' becomes 'd' (musical note).
'fancy' indicates anagramming the letters.
'their' anagrammed gives 'ither'.
'd'+'ither'='DITHER'
(Other definitions for dither that I've seen before include "quake" , "State of high agitation" , "Be undecided and somewhat agitated" , "Faff about" , "indecision" .)