The allegations were only the latest in a wide-ranging federal corruption investigation that has already led to indictments not only against Mr. Mangano, a Republican, but also against his wife, Linda Mangano, and the Town of Oyster Bay, N.Y., supervisor, John Venditto, a Republican as well. All three have been charged with taking part in a scheme that included bribery, kickbacks and a no-show job involving food tasting. Their trial is scheduled to begin in a few weeks.

The indictments, brought by the United States attorney’s office for the Eastern District of New York, which oversees Brooklyn and Long Island, have sent ripples through the political sinew of Nassau County, where Mr. Mangano once wielded influence through his warm ties to the Republican establishment and to Gov. Andrew M. Cuomo, a Democrat.

According to court papers, Mr. Walker, who is known as Rob, spoke with the contractor several times last year in an effort to persuade him to conceal the $5,000 exchange from a federal grand jury, urging him at times to lie about “both the character of the payment and the reasons for the payment.” The contractor had given Mr. Walker the money in 2014, prosecutors say, when they went together to Indiana to attend a University of Notre Dame football game.

While he was under investigation, Mr. Walker testified as a prosecution witness at the federal corruption trial of Dean G. Skelos, the former leader of the State Senate — and another Nassau County Republican — who was convicted with his son, Adam B. Skelos, in 2015 of bribery, extortion and conspiracy. Last September, the convictions of both men were tossed out by a federal appeals court that cited a United States Supreme Court decision that had narrowed the legal definition of corruption.

Speaking to reporters after Mr. Walker’s arraignment, his lawyer, Brian Griffin, said that the government had investigated his client for two and a half years and that the charges against him were “not a public corruption indictment.” They emerged instead, Mr. Griffin said, from “a personal trip with a 20-year friend.”