Pudding’s one conveniently run up with South American tuber (7)

I believe the answer is:

tapioca

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'pudding's' is the definition.
(tapioca pudding is a kind of pudding)

I cannot understand how the rest of the clue works.

Can you help me to learn more?

(Other definitions for tapioca that I've seen before include "Foodstuff used as a thickener" , "Starchy white substance used to make puddings" , "Starch from cassava used in making puddings" , "course" , "Cassava product used for puddings" .)

I've seen this clue in The Times.

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