New York’s advertising campaigns promoting the state’s economic-development opportunities have faced criticism for their high costs and meager returns in job creation. Now they have encountered another challenge.

A former North Carolina Supreme Court judge has filed complaints with the U.S. Office of Special Counsel and a New York state ethics panel alleging that taxpayer-funded ads Gov. Andrew Cuomo’s administration has run in North Carolina violated the Hatch Act by interfering with the gubernatorial and legislative races in that state.

The commercials, which reference the state’s so-called transgender bathroom law, among policies by other states, “mention North Carolina and its leadership in a transparent attempt to criticize, interfere and affect the impending North Carolina elections,” according to the federal complaint filed by the retired judge, Robert F. Orr.

The Republican governor of North Carolina, Pat McCrory, is up for re-election this year, as is the state’s legislature. The Hatch Act prohibits federal, state and local employees from using their official authority to interfere with or influence the outcome of an election or nomination for office.

A spokesman for Mr. Cuomo, John P.L. Kelly, described the complaints as “frivolous” but didn’t address the allegation concerning any attempt to sway North Carolina’s elections.

“It would be funny if the issues we were actually talking about did not involve human rights and basic equality,” Mr. Kelly said. “New York welcomes everyone—everyone—who seeks to embrace freedom while pursuing their dreams.”

Mr. Orr, who said in an interview that he isn’t affiliated with the North Carolina governor’s administration or campaign but has done some legal work for the state, has filed a similar complaint with New York’s Joint Commission on Public Ethics.

The ads, which the complaints say began running in North Carolina in late June, promote New York’s business climate as “where the true leaders are” and “where we understand the value of diversity.” The ads are no longer airing, according to Mr. Cuomo’s office.

The federal complaint criticizes the ads as going “beyond appropriate economic-development recruitment,” saying: “By using public funds to promote New York as supporting certain policies and implicitly criticizing contrary political decisions made in North Carolina, an ethical imitation has been breached.”

The complaint specifically identifies Mr. Cuomo as having violated the Hatch Act, along with several employees of Empire State Development, New York’s economic-development agency.

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