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Movement in water of peacock, say, by river in Staffs city (9,6)

Ross

I believe the answer is:

butterfly stroke

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'movement in water' is the definition.
(swimming stroke)

'peacock say by river in staffs city' is the wordplay.
'peacock say' becomes 'butterfly' (peacock is a kind of butterfly).
'by river' becomes 'r' (abbreviation. I am not sure about the 'by' bit.).
'in' means one lot of letters goes inside another.
'staffs city' becomes 'Stoke' (Stoke-on-Trent in Staffordshire).
'r' inserted into 'stoke' is 'stroke'.
'butterfly'+'stroke'='BUTTERFLY STROKE'

'of' is the link.

Can you help me to learn more?

(Another definition for butterfly stroke that I've seen is " Swimming style".)

I've seen this clue in The Times.
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