The part that hurt (3)

I believe the answer is:

bit

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'the part' is the definition.
(both mean a piece of something)

I cannot understand how the remainder of the clue works.

Can you help me to learn more?

(Other definitions for bit that I've seen before include "Unit of information" , "Segment" , "something boring" , "a curb?" , "Small piece; boring tool" .)

I've seen this clue in the Evening Standard.

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