Hostility upset period around New Year (6)
I believe the answer is:
enmity
'hostility' is the definition.
(synonyms)
'upset period around new year' is the wordplay.
'upset period' becomes 'emit' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'around' is an insertion indicator.
'new' becomes 'n' (abbreviation as in New Testament NT).
'year' becomes 'y'.
'emit' placed around 'n' is 'enmit'.
'enmit'+'y'='ENMITY'
Can you help me to learn more?
(Other definitions for enmity that I've seen before include "discord" , "Bad blood" , "needle" , "ill will" , "Antagonism, ill-will" .)