The mayor's response tells you a lot about the authority he holds. If the dentist sends the bill to the town, the town will pay for it. If the dentist sends the bill to the mayor, apparently the town will also pay for it. The mayor's response shows us that the town will pay all of his expenses, which gives us insight into the level of power he has (a high level).
The response also shows the mayor's attitude at this point in the story. His refusal to look at the dentist when he says it, and his use of the word "damn," show that he is not necessarily pleased to admit the truth to the dentist. So he knows he has absolute authority, and perhaps he admits some guilt in this situation, but he is resigned to it.