Unwell after stench from pit comes to town (7)

I believe the answer is:

borough

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'town' is the definition.
(borough is a kind of town)

I cannot understand how the remainder of the clue works.

(Other definitions for borough that I've seen before include "Administrative area" , "Region of London" , "London district" , "Local district" , "Important town" .)

I've seen this clue in The Telegraph.

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