Ill-humour comes out when one contests supremacy (8)

I believe the answer is:

competes

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'ill-humour' is the definition.
I can't tell whether this definition defines the answer.

'comes out when one contests supremacy' is the wordplay.
'out' is an insertion indicator.
'when one contests supremacy' becomes 'pet' (I can't explain this - if you can you should believe this answer much more).
'comes' going around 'pet' is 'COMPETES'.

Can you help me to learn more?

(Other definitions for competes that I've seen before include "Contends for a prize" , "Vies" .)

I've seen this clue in the Evening Standard.

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