Lift the black soot out (5)
I believe the answer is:
boost
'lift' is the definition.
(I know that boost is a more specific form of the action lift)
'black soot out' is the wordplay.
'black' becomes 'b' (as in pencil classifications).
'out' indicates anagramming the letters (out can mean wrong or inaccurate).
'soot' anagrammed gives 'oost'.
'b'+'oost'='BOOST'
'the' is the link.
(Other definitions for boost that I've seen before include "Encourage or increase" , "Help to increase" , "Push upwards, help" , "lift" , "Improve - pick-me-up" .)