How did many U.S. cities try to increase government efficiency and lessen corruption?
The major way that cities tried to do this during the Progressive Era was by trying to destroy the political "machines" that ran the cities.
One of the main ways in which the cities tried to do this was by reforming their electoral systems. They instituted things like at-large elections and nonpartisan elections to try to reduce the power of the political parties and of various ethnic groups that were concentrated in small parts of the cities. This was meant to lead to more efficiency and less corruption because it was believed that people elected in these sorts of elections would be good government types who were running to help the city rather than to enrich themselves as part of a political machine. They thought they would get good, middle-class people who were trying to work for the "greater good."