“We are preparing badly for the future,” said Chiara Saraceno, an Italian sociologist who works on education.

Italy’s prime minister, Mario Draghi, allowed all Italian high school students to go back to school in person for at least half of their classes starting on Monday. Finishing the academic year in class, Mr. Draghi has said, should be a priority.

“The whole government thinks that school is a fundamental backbone of our society,” said Italy’s health minister, Roberto Speranza. “The first place where we will invest.”

But a good deal of damage has already been done.

Throughout much of the last year, the government argued that keeping high schools closed was necessary to prevent infection on the public transportation that students took to and from class.

Elementary schools were allowed to open more often, but the country’s insistence on closures, especially of middle and high schools, experts say, risked exacerbating inequalities and the country’s profound north-south divide. National and regional officials drew sharp criticism, and even the education minister who was in office then argued that schools should have opened more.

Mr. Speranza acknowledged that schools had paid “a very high price in these months.”

Schools around the southern city of Naples have remained closed longer than the rest of the country, in part because the president of the Campania region, Vincenzo De Luca, insisted they were a potential source of infection. At one point, he mocked the notion that children in his region were “crying to go to school.”

In Naples, the dropout rate is about 20 percent, twice the European average, and in the city’s outskirts it is even higher. Teachers there have struggled to keep students interested in school, and worry that months of closed classrooms would shut students out for good.

As schools closed Francesco Saturno, 13, spent his mornings helping in his grandfather’s fruit shop, sleeping in or glued to his PlayStation. He only twice logged on to his online class.

His mother, Angela Esposito, 33, who herself dropped out of high school, worried that he might leave school and follow in the footsteps of his father, who earns tips of loose change for babysitting parked cars in Naples.

“I am scared that if he doesn’t go to school he is going to get lost,” she said. “And getting lost in Naples is dangerous.”

In Italy, it is illegal for students below the age of 16 to drop out of school, and the local prosecutor for the minors’ court, aware that social workers are swamped, asked school principals to report dropout cases directly to her.

“I am really worried,” said the prosecutor, Maria De Luzenberger. In the last month, about a thousand drop out cases from Naples and the nearby city of Caserta have piled up on her desk, she said. That was more than in all of 2019. “I didn’t expect such a flood.”

Colomba Punzo, the principal of Francesco’s school, said dropouts had tripled in her primary and middle school during the school closures. She scrambled to find an alternative, and organized in-person workshops every morning to get Francesco and other at-risk children back into the system.

Ms. Punzo said policymakers underestimated how closing schools in neighborhoods like Ponticelli meant cutting “the only possible lifeline,” for the children. “When the school is open you can grab them and make them come, when the school is closed what do you do?”

In Naples’ Scampia district, known across Italy as a tough place plagued for years by the Camorra mafia, teachers at the Melissa Bassi High School had made significant progress in getting local children into school through art projects, workshops and personal tutoring.

The school’s principal said half of its students stopped following classes when they moved online. He said they gave cellphone SIM cards to those who could not afford Wi-Fi and offered evening lessons to teenagers forced to work as the pandemic hit their families’ finances.

But the challenge was enormous. Some of the neighborhood’s most neglected housing projects lack cellphone coverage, and children are often crammed with multiple family members into a few rooms. Teachers hoped most of the students would return if and when schools reopened, but they feared those who fell behind won’t see the point of going back.

“They are so discouraged,” said Marta Compagnone, a teacher there. “They think the bets are off.”

Hanging out with his friends on the steps of a square below the “Sails,” a huge triangular housing project a few blocks from Melissa Bassi High School, Giordano Francesco, 16, said he often fell asleep, grew bored and frustrated with the online classes he followed on his phone. He got into arguments with teachers because he often logged off to help his grandfather, who has Alzheimer’s disease, eat or use the bathroom.

His mother, who left school at 10 and lost her job as a theater cleaner during the pandemic, asked him to finish the school year. He said he would, and then drop out afterward.

His girlfriend, Marika Iorio, 15, standing next to him, said she intended to graduate, become a psychologist and live a different life from her father, who cannot read or write. But she was struggling to follow school online and failing her classes, too.

“I am scared I might not make it,” she said.