Cotton on small piece of wood (4)
I believe the answer is:
twig
This clue is a double definition.
'cotton on' is the first definition.
(I know this)
'small piece of wood' is the second definition.
(I've seen this before)
(Other definitions for twig that I've seen before include "Get the idea" , "bit of wood" , "Suddenly see - bit of a tree" , "Suddenly comprehend" , "Only a minor branch" .)