CENTRAL ISLIP, N.Y. — When a New York restaurateur who had donated to Mayor Bill de Blasio’s campaign grew increasingly desperate to get a lease renewed, he sent an email directly to one of the city’s most powerful people: Emma Wolfe, a top aide to the mayor.

“Hope you had a great holiday season,” the restaurateur, Harendra Singh, wrote in January 2015. “I would love to speak with you at your convenience.”

Ms. Wolfe took just nine minutes to respond: “Absolutely,” she wrote, “let’s talk on Monday.”

A meeting was soon arranged — part of a longstanding effort that federal prosecutors say Mr. de Blasio and his aides undertook to help Mr. Singh in return for campaign contributions.

The emails were among a raft of evidence related to the alleged scheme that was revealed in Federal District Court here on Wednesday, as part of a case that has revived discussions of possible impropriety by Mr. de Blasio, despite his not actually being involved in the trial.