Have whittled fragments — that’s annoying (4,3,5)

I believe the answer is:

what the devil

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'annoying' is the definition.
I don't know anything about this answer so I cannot tell whether this works.

'have whittled fragments' is the wordplay.
'fragments' indicates anagramming the letters (the fragments are put back together differently).
'have'+'whittled'='havewhittled'
'havewhittled' is an anagram of 'WHAT THE DEVIL'.

'that's' acts as a link.

Can you help me to learn more?

(Another definition for what the devil that I've seen is " emphatic introduction to question?".)

I've seen this clue in The Times.

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