Walked and, for a little way, rode (6)

I believe the answer is:

strode

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'walked' is the definition.
(I've seen this before)

'way rode' is the wordplay.
'way' becomes 'st' (abbreviation for street).
'st'+'rode'='STRODE'

'and for a little' is the link.
I am not very happy about this link. Some or all of it may belong to another bit of the clue.

Can you help me to learn more?

(Other definitions for strode that I've seen before include "Galumphed" , "went on foot" , "Paced" , "Walked fast with long steps" .)

I've seen this clue in the Evening Standard.

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