Firstly, bad arborists rarely keep this part of the tree (4)
I believe the answer is:
bark
'part of the tree' is the definition.
('bark' is part of 'tree')
'firstly bad arborists rarely keep' is the wordplay.
'firstly' suggests taking the first letters.
The first letters of 'bad arborists rarely keep' is 'BARK'.
'this' acts as a link.
(Other definitions for bark that I've seen before include "Ship" , "Bay" , "Shout gruffly; part of tree" , "Sound of dog - protector of trunks" , "Outside of a tree" .)