Timely investigations might not have uncovered now-fired Department of Education staffer David Hay’s alleged pedophilia, but his case exposes some pretty major holes in the city’s vetting of new hires.

A proper check might’ve uncovered the unethical behavior that prompted him to quit a high-school post in his native Wisconsin. But that isn’t the ball the city dropped when it came to Hay.

On Sunday, The Post’s Susan Edelman reported charges that a city investigation of Hay’s bullying use of social media was shelved to avoid bad press for Mayor Bill de Blasio during his failed White House run. It was one of nine blocked probes, according to a whistleblower letter sent to three city councilmen last August.

Soon after his arrest in a Wisconsin pedophile sting, city Department of Investigation Commissioner Margaret Garnett explained that Hay’s case was one of some “6,000 backlogged background files.” But she also noted that DOI’s check is meant to enhance the hiring processes of other agencies.

Hay underwent two DOE background checks, when he was hired in 2016 and promoted in 2018. But it seems neither included any serious work-history vetting.

After Hay’s arrest, the Milwaukee Journal Sentinel reported that he had quit his post at Kettle Moraine HS in Milwaukee years earlier after being caught improperly using a district credit card for personal charges, as well as lacking the license needed for that job. He got another high-school post before going to Harvard in 2014 to earn a doctorate, after which the city DOE hired him — without making enough calls.

Then, when he got into trouble last year for his intemperate tweeting (in defense of Chancellor Richard Carranza, as it happens) while IDing himself as a DOE official, city officials opted to ignore (or bury) the warning signs.

Did the supposedly-independent schools Special Commissioner of Investigations punt to keep City Hall happy? New Yorkers need to know. And the DOE and other agencies need to stop leaving it to the DOI to do the real checking.