Groan, perhaps, having a change of heart? (5,10)

I believe the answer is:

organ transplant

I'm a little stuck... Click here to teach me more about this clue!

'change of heart?' is the definition.
(I've seen this in another clue)

'groan perhaps having a' is the wordplay.
I cannot really understand how this works, but
an anagram of 'groan' is 'organ' which is present in the answer.
'a' could be 'an' and 'an' is located in the answer.
This explanation may well be incorrect...

Can you help me to learn more?

I've seen this clue in The Independent.

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