Patrick says hi in Italian and cooks them pudding (7)

I believe the answer is:

tapioca

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

I can't explain the remainder of the clue.

(Other definitions for tapioca that I've seen before include "Sago pudding" , "Foodstuff used as a thickener" , "Pudding maker" , "course" , "Starchy foodstuff" .)

I've seen this clue in The Irish Times.

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