Treat core illness in clerical office (9)

I believe the answer is:

rectorate

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'clerical office' is the definition.
I don't know anything about this answer so I can't tell whether it can be defined by this definition.

'treat core illness' is the wordplay.
'treat core' anagrams to 'RECTORATE'.
However, I am not clear how this is indicated.

'in' acts as a link.

Can you help me to learn more?

I've seen this clue in the Sydney Morning Herald.

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