Tearing up my carols, he improvised (10)

I believe the answer is:

lachrymose

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'tearing up' is the definition.
The definition and answer are not the same part of speech.

'my carols he improvised' is the wordplay.
'improvised' indicates anagramming the letters (letters are improvised or randomised).
'my'+'carols'+'he'='mycarolshe'
'mycarolshe' with letters rearranged gives 'LACHRYMOSE'.

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(Other definitions for lachrymose that I've seen before include "Weepy" , "This is given to crying" , "In bits" , "showing sorrow" , "Sad" .)

I've seen this clue in The Times and The Sunday Times.

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