Ms. Garnett said she inherited a massive backlog of 6,000 unfinished background checks when she took control of the department in January, some dating as far back as 2015. “The risks presented by this example are exactly why I took immediate steps to assess and then reorganize the Background Investigation Unit,” she said, referring to Mr. Hay.

The scope of the problem is still unclear, though the unfinished background checks apply to high-ranking officials and policymakers across Mayor Bill de Blasio’s administration. Ms. Garnett said she had taken immediate action to begin to address the issue.

Officer Zuehls, of Neenah, declined to offer more information about the specific charges or why Mr. Hay was arrested at an airport.

Wisconsin state law indicates that someone accused of using a computer to facilitate a child sex crime is suspected of attempting to sexually abuse a child. The law defines that charge as a person who “uses a computerized communication system to communicate with an individual who the actor believes or has reason to believe has not attained the age of 16 years with intent to have sexual contact or sexual intercourse with the individual.”

It is not yet clear when the investigation into Mr. Hay was launched, or whether Mr. Hay has obtained a lawyer.

The investigation was conducted by the Neenah police department, with help from the Milwaukee County Sheriff’s Department, though it is not yet clear what precise role the Milwaukee police played.

The investigation is ongoing, and there is no information yet about when Mr. Hay will be arraigned.

Mr. Hay grew up in the small town of Antigo, Wis., about three hours from Milwaukee, according to a 2017 interview.