Old king repeatedly can start to grouch, showing pique (3-7)

I believe the answer is:

tut-tutting

'showing pique' is the definition.
(tutting shows irritation or pique)

'old king repeatedly can start to grouch' is the wordplay.
'old king' becomes 'tut' (King Tut).
'repeatedly' says to double the letters.
'can' becomes 'tin' (synonyms).
'start to' suggests taking the first letters.
The initial letter of 'grouch' is 'g'.
'tut' doubled is 'tuttut'.
'tuttut'+'tin'+'g'='TUT-TUTTING'

I've seen this clue in The Times.

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