Maybe John's pet, nursed by vet, is sick (10)

I believe the answer is:

evangelist

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'maybe john's' is the definition.
I can't tell whether this definition defines the answer.

'pet nursed by vet is sick' is the wordplay.
'pet' becomes 'angel' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'nursed by' indicates putting letters inside.
'sick' indicates anagramming the letters.
'vet'+'is'='vetis'
'vetis' is an anagram of 'evist'.
'angel' going inside 'evist' is 'EVANGELIST'.

Can you help me to learn more?

(Other definitions for evangelist that I've seen before include "Mark for one" , "Preacher" , "Gospel spreader" , "Gospel writer" , "Zealous believer" .)

I've seen this clue in The Independent.

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