Put up with a big-headed listener (4)

I believe the answer is:

bear

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'put up with' is the definition.
(synonyms)

'a big-headed listener' is the wordplay.
'a big-headed' becomes 'b' (this might be a standard abbreviation which I don't know about).
'listener' becomes 'ear' (body part which hears or listens).
'b'+'ear'='BEAR'

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(Other definitions for bear that I've seen before include "someone being ill-mannered" , "Support (a weight)" , "Tolerate - carnivore" , "Relating to charged particles" , "Carry - teddy" .)

I've seen this clue in the Evening Standard.

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