Peter is thus made peer in a demoralising way (15)

I believe the answer is:

dishearteningly

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'peter is thus made peer' is the definition.
I can't judge whether this defines the answer.

'demoralising way' is the wordplay.
'demoralising' becomes 'disheartening' (synonyms).
'way' becomes 'ly' (I am not sure about this - if you are sure you should believe this answer much more).
'disheartening'+'ly'='DISHEARTENINGLY'

'in a' acts as a link.

Can you help me to learn more?

I've seen this clue in The Guardian.

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