Disheartened Italian fan hit priest? That's not right (10)

I believe the answer is:

inaccurate

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'not right' is the definition.
(I've seen this before)

'disheartened italian fan hit priest?' is the wordplay.
'disheartened' means to remove the middle letters (the middle or heart is removed).
'fan hit' becomes 'ac' (I can't justify this - if you can you should give a lot more credence to this answer).
'priest?' becomes 'curate' (I've seen this before).
'italian' with its middle taken out is 'in'.
'in'+'ac'+'curate'='INACCURATE'

'that's' is the link.

Can you help me to learn more?

(Other definitions for inaccurate that I've seen before include "not hitting all the right notes?" , "wrong" , "Unreliable" .)

I've seen this clue in The Irish Times.

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