At Richmond’s Southside Community Center, slated to balloon in capacity with the help of rescue plan funding, Linda Scott, a 75-year-old pickleball enthusiast, said she had heard nothing of the coming upgrades.

“I know that we’re getting lots of money,” said Ms. Scott, a self-described independent who voted for Mr. Biden. “But what we’re doing with it, I’m not sure.”

Thirteen months after Mr. Biden signed the emergency package, that money is starting to fuel a wave of investment on city infrastructure, public services and pilot programs unlike any in decades…

Mr. Stoney allowed that it was not clear how much voters had processed that barrage of spending when the projects were far from completion. In cities like his, the money must make its way through city councils and contract-bidding processes; in some states, the path to deploying funds has been even longer as governors wrangle with conservative legislatures.