Sourly referring to a department duly halved (6)
I believe the answer is:
acidly
'sourly referring' is the definition.
I can't tell whether this defines the answer.
'a department duly halved' is the wordplay.
'department' becomes 'cid' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'halved' means to take half the letters.
'duly' cut in half is 'ly'.
'a'+'cid'+'ly'='ACIDLY'
'to' acts as a link.
Can you help me to learn more?
(Other definitions for acidly that I've seen before include "in caustic manner" , "in a sour manner" , "Bitterly" .)