Initially appear bitter about cutting sick tree's root out (9)
I believe the answer is:
eradicate
'root out' is the definition.
(root out can mean to eradicate or remove)
'initially appear bitter about cutting sick tree's' is the wordplay.
'initially' suggests taking the first letters.
'bitter' becomes 'acid' (acid can mean bitter-tasting).
'about' says the letters should be written in reverse.
'cutting' means one lot of letters goes inside another (some letters cut their way into a word).
'sick' indicates anagramming the letters.
The first letter of 'appear' is 'a'.
'acid' written backwards gives 'dica'.
'a'+'dica'='adica'
'tree' anagrammed gives 'erte'.
'adica' inserted inside 'erte' is 'ERADICATE'.
(Other definitions for eradicate that I've seen before include "Expunge" , "Excise" , "Get rid of, remove all traces" , "scratch" , "Extinguish, wipe out" .)