Hostile spirit serves as an impetus to get rid of favourite (6)
I believe the answer is:
animus
'hostile spirit' is the definition.
'animus' can be an answer for 'spirit' (I have seen 'spirit of hostility' mean 'animus' so perhaps 'spirit' could also mean 'animus'). I am unsure of the 'hostile' bit.
'an impetus to get rid of favourite' is the wordplay.
'to get rid of' indicates named letters should be taken away.
'favourite' becomes 'pet' (synonyms).
'impetus' with 'pet' taken away is 'imus'.
'an'+'imus'='ANIMUS'
'serves as' is the link.
Can you help me to learn more?
(Other definitions for animus that I've seen before include "Bad blood" , "A feeling of ill will or hostility" , "Hostility, ill-feeling" , "Hostile feeling" , "Motivating spirit" .)