The given sentence contains a pronoun agreement error as 'many'(plural indefinite pronoun) is wrongly placed to indicate the free choice. It needs to be replaced with 'any' which is grammatically correct to be used for denoting a free choice irrespective of gender or number. It is used for both as a singular noun and also for the plural and uncountable nouns(in the given case) and thus, it will be followed by the antecedent 'they' and the correct sentence reads as:
"Whenever any of the mayor's advisors meet in Joe West's treehouse, they finish all their work."
