The shame of a police inspector's girl (8)

I believe the answer is:

disgrace

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'the shame' is the definition.
(synonyms)

'inspector's girl' is the wordplay.
'inspector's' becomes 'di's' (abbreviation for Detective Inspector).
'girl' becomes 'grace' ().
'dis'+'grace'='DISGRACE'

'of a police' acts as a link.
This may not be correct. Some or all of it may be part of another bit of the clue.

Can you help me to learn more?

(Other definitions for disgrace that I've seen before include "Loss of honour" , "Loss of reputation as the result of dishonourable action" , "Shame or loss of respect" , "scandalous" , "Cadger is in state of ignominy" .)

I've seen this clue in the Evening Standard.

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