Cheek, or something next to it! (3)

I believe the answer is:

lip

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'cheek' is the definition.
(both can mean insolence or rudeness)

I cannot understand how the rest of the clue works.

(Other definitions for lip that I've seen before include "Rim; impertinence" , "Insolence (slang)" , "Impudent talk from part of the face" , "Aural impudence" , "giving such insincere service" .)

I've seen this clue in The Times.

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