Prior to the latest vision, the city’s last major broadband intervention was negotiated under Mayor Michael R. Bloomberg in 2006. New York signed a franchise agreement with Verizon that gave the company the privilege of burying fiber optic cables under the city streets in exchange for installing high-end cable-speed fios in every neighborhood. But Verizon has failed to do this in many low-income neighborhoods. In an April public hearing, the city’s chief technology officer, John Paul Farmer, said that with relatively few vendors in some neighborhoods, there was little pressure on the market to cut prices. “The current oligopolistic system is broken and has brought digital inequality to the streets and neighborhoods of New York,” he said.

The city recently reached an agreement with Verizon requiring an additional 500,000 homes to be connected by 2023, including at least 125,000 in underserved neighborhoods.

Chris Serico, a Verizon spokesman, said the company was on track to meet the terms of the settlement. “Verizon is committed to finding long-term solutions that will provide affordable broadband options to low-income Americans,” Serico wrote in an email.

Clayton Banks, the CEO of Silicon Harlem, a company focused on improving connectivity in Harlem, said he hoped the city’s strategy of betting on more competition would work, but he was waiting, like Fios and the current providers would look priced. “If you continue to build the infrastructure, which is certainly welcome and necessary, but keep the retail price,” he said, “you haven’t solved anything to get more people online.”

After months of back and forth, NYC Mesh got the go-ahead to join two other developments in the Bronx and Queens to build a hub on the 24-story council housing tower in Bed-Stuy. Four other small providers, including Silicon Harlem, were selected to wire 10 more NYCHA developments. As part of the first phase of the Internet master plan, for which the city is allocating $ 157 million, NYC Mesh installed free public hotspots around the outside of the projects; the other companies are required to give residents access to Wi-Fi in their homes for no more than $ 20 a month.

NYC Mesh has applied to set up hubs in an additional 163 public buildings as part of the second phase. If successful, NYC Mesh could cover much of the city in the next five to seven years. Since every router installation comes with a free public WiFi hotspot, NYC Mesh could help make the Internet truly universal across New York City.

Even if NYC Mesh has grown continuously, it still has the same problems as the big providers: the internet sometimes goes off. Mr. Heredia and other volunteers take pride in resolving service issues quickly, but as the organization grows it needs more people like Mr. Heredia if it is to keep members happy.