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The generic British word for dessert is "pudding."
In the 19th century, the "g" was sometimes pronounced as a harder "k." Sometimes, the "n" got dropped. Sometimes that was shortened by slicing off the "pud."
In other words, small, incremental changes resulted in pudding->puddink->puddik->dick.
It's not meant to be dirty; it's just a Victorian synonym for "dessert."
Pepper a cake with currants or raisins, and you get "spots" in your pudding, hence: spotted dick.