Wild tarragon is put down heartlessly in a self-important way (10)

I believe the answer is:

arrogantly

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

'in a self-important way' is the definition.
I can't tell whether this definition defines the answer.

'wild tarragon is put down heartlessly' is the wordplay.
'wild' indicates an anagram.
'is put down' becomes 'lay' (I've seen this before).
'heartlessly' suggests removing the centre.
'lay' with its middle taken out is 'ly'.
'tarragon' is an anagram of 'arrogant'.
'arrogant'+'ly'='ARROGANTLY'

Can you help me to learn more?

(Another definition for arrogantly that I've seen is " with haughtiness".)

I've seen this clue in the Irish Times.

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