Part of a cigarette end that goes in your mouth! (4)

I believe the answer is:

bite

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This clue could be a double definition.

'part of a cigarette end that goes in your' is the first definition.
'bite' can be an answer for 'part' (thesaurus). I'm not sure about the remainder of the definition.

'mouth' is the second definition.
(thesaurus)

Can you help me to learn more?

(Other definitions for bite that I've seen before include "Light meal" , "Puncture by 9 [ADDER]" , "Take effect; snack" , "champ" , "Use one's teeth" .)

I've seen this clue in the Evening Standard.

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