Adam leaves chambermaids to drag through the mud (8)

I believe the answer is:

besmirch

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

'mud' is the definition.
The definition and answer can be both to do with contact as well as being verbs in their base form.
Maybe there's a link between them I don't understand?

I don't understand how the remainder of the clue works.

Can you help me to learn more?

(Other definitions for besmirch that I've seen before include "Sully" , "Foul" , "run down" , "soil" , "dull" .)

I've seen this clue in the Irish Times and the Irish Times (Crosaire).

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