Bank broken by woman without thinking (8)

I believe the answer is:

remissly

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'without thinking' is the definition.
Both the definition and answer are adverbs.
Maybe you can see a link between them that I don't see?

'bank broken by woman' is the wordplay.
'bank' becomes 'rely' (to bank on is to rely on).
'broken by' indicates putting letters inside.
'woman' becomes 'miss' (miss is a kind of woman).
'rely' going around 'miss' is 'REMISSLY'.

Can you help me to learn more?

(Another definition for remissly that I've seen is " with slipshod attitude".)

I've seen this clue in The Sunday Times.

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