Red mitten upset friend in a hurtful way (13)

I believe the answer is:

detrimentally

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'in a hurtful way' is the definition.
I can't judge whether this definition defines the answer.

'red mitten upset friend' is the wordplay.
'upset' indicates an anagram.
'friend' becomes 'ally' (an ally is a friend or supporter).
'red'+'mitten'='redmitten'
'redmitten' is an anagram of 'detriment'.
'detriment'+'ally'='DETRIMENTALLY'

Can you help me to learn more?

(Another definition for detrimentally that I've seen is " In a damaging manner".)

I've seen this clue in The Daily Express.

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