Old Bill's beginnings of a beard? (4)

I believe the answer is:

fuzz

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'beginnings of a beard?' is the definition.
(I've seen this in another clue)

I don't understand how the remainder of the clue works.

Can you help me to learn more?

(Other definitions for fuzz that I've seen before include "Frizzy hair; blurred image" , "Police - fluff" , "Old Bill (slang)" , "Met, perhaps" , "Hairy layer" .)

I've seen this clue in The Sun.

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