Ten kids died despite each being the subject of at least four abuse or maltreatment complaints to the city’s troubled child welfare agency in the weeks leading up to the slaying of little Zymere Perkins, a probe by Comptroller Scott Stringer’s office has found.

The tragedies were among 38 deaths involving vulnerable children flagged for high-priority investigations by the Administration for Children’s Services, Stringer found in the report, a copy of which was obtained by The Post.

In a letter to outgoing ACS Commissioner Gladys Carrion — who announced she was stepping down last week amid scrutiny of her handling of the agency — Stringer said his findings “starkly illustrate ACS’ persistent lack of progress in meeting its own targets for how those investigations are conducted, supervised and managed.”

Without reform, Stringer warned, the bungling “will continue to prove fatal for an unknown number of children who will foreseeably need to rely on ACS for protection from abusive individuals in their own households.”

Carrion’s abrupt retirement came one day before the state ordered the appointment of an outside monitor to oversee ACS due to shocking lapses in the handling of Zymere’s case.

Stringer’s report, which will be made public Thursday, said his office looked into 3,692 high-priority ACS probes opened between July 1 and Sept. 25 — one day before Zymere, 6, was beaten to death in Harlem following five ACS investigations into abuse allegations against his family.

Zymere’s mom, Geraldine Perkins, and her boyfriend, Rysheim Smith, have been charged with endangering the welfare of a child. Authorities suspect Smith fatally bludgeoned the child with a broomstick.

The comptroller’s probe found that 53 high-priority cases involving four or more complaints against a household were closed without any agency employees having “face-to-face contact” with the children in question.

Stringer also uncovered multiple instances in which ACS failed to follow official protocol in handling the investigations. The lapses included 22.4 percent of the 3,692 cases in which there was no “face-to-face contact” with the child within 24 hours, and 25.9 percent that were closed without the “requisite number of face-to-face contacts with the child” every other week.

In addition, 31.9 percent of the cases were closed without first being reviewed by a supervisor five times, as required, and 72.5 percent were closed without being reviewed by a manager even three times.

Those failures occurred despite ACS being so top-heavy with management that supervisors actually outnumber the caseworkers assigned to deal with kids at risk, as revealed by The Post on Monday.

ACS also failed to produce a risk-assessment profile within 40 days, as required, in 68 percent of the cases, according to Stringer’s preliminary findings.

“It’s so alarming and concerning,” Stringer told The Post. “The sheer number of kids who have tragically slipped through the cracks makes your heart break.”

Since Zymere’s death, another child, 3-year-old Jaden Jordan, died from a beating, allegedly administered by his mother’s boyfriend, Salvatore Lucchesse.

That abuse was uncovered on Nov. 28, three days after ACS workers investigated a tip that mistakenly gave the wrong address for the family’s home in the Gravesend section of Brooklyn.

In his letter, Stringer said the probe was a follow-up to a June 15 report that “found significant deficiencies in ACS’s processes.”

In addition to Stringer’s findings, ACS is bracing for the release of a report by the city Department of Investigation early next year. Last week, DOI Commissioner Mark Peters said his probe would reveal “failures at the highest level” of ACS.

Both ACS and Mayor Bill de Blasio’s office disputed Stringer’s findings, saying 21 of the 38 deaths Stringer cited had no prior history with ACS.

They also said seven other deaths stemmed from unsafe sleeping conditions, two were illness-related, one was determined to be accidental, and the official cause of two others remained pending.

The existence of five other child death cases cited by Stringer is being disputed by the de Blasio administration.

“It’s no surprise that ACS’s data was cherry-picked to support a simplified and largely inaccurate conclusion,” mayoral spokesperson Aja Worthy-Davis said.

“This report contains many inaccuracies — such as a base misunderstanding of child-protective review protocols and legal rules regarding risk assessment.

“Our focus remains on aggressive reform meant to protect every child that we interact with.”

Additional reporting by Bruce Golding