This person’s outstanding, cutting old woman’s shrub (6)
I believe the answer is:
mimosa
'shrub' is the definition.
(mimosa is a kind of shrub)
'this person's outstanding cutting old woman's' is the wordplay.
'this person's' becomes 'I'm' (('I am' - 'this person' means the crossword setter).
'outstanding' becomes 'os' (accounting abbreviation).
'cutting' is an insertion indicator (some letters cut their way into a word).
'old woman' becomes 'ma' (both can mean someone's mother).
'im'+'os'='imos'
'imos' placed inside 'ma' is 'MIMOSA'.
(Other definitions for mimosa that I've seen before include "cocktail" , "So maim with tropical shrub bearing fragrant flowers" , "Shrub - I'm Amos (anag)" , "Australian acacia tree" , "Sensitive tree or shrub with globular clusters of yellow flowers" .)