Cotton on branch (4)
I believe the answer is:
twig
I believe this is a double definition.
'cotton on' is the first definition.
('twig' can be a synonym of 'cotton on')
'branch' is the second definition.
(twig is a kind of branch)
(Other definitions for twig that I've seen before include "bit of stick" , "Small woody shoot" , "Finally understand" , "Tiny end of branch" , "Suddenly realise" .)