When one of New York City’s most prestigious private schools hired Winston Nguyen in 2020, administrators knew about the felony conviction for fraud in his troubled past. But the second chance they offered him backfired. Nearly four years later, Mr. Nguyen, a math teacher, was arrested again, accused of preying on students. And the school, Saint Ann’s in Brooklyn, faced a roiling crisis.

On Monday, Mr. Nguyen, 38, pleaded guilty to a felony and several misdemeanors after being charged with soliciting lewd images and videos from students. When he is sentenced, most likely later this month, he faces a possible seven-year prison term.

His plea marks the latest chapter in a scandal that has marred the reputation of Saint Ann’s School and the administrators who hired him.

This is the second time Mr. Nguyen has been convicted of a felony. In 2019, he pleaded guilty to grand larceny and other charges after he was accused of stealing more than $300,000 from his employers, an older couple he worked for as a home health aide.

He served four months at the Rikers Island jail complex, and about a year later was hired by Saint Ann’s, a school that charges about $60,000 per year in tuition and caters to New York’s wealthy creative class.

A nattily dressed figure who arrived at class often in a suit and sometimes with a bow tie, Mr. Nguyen transformed a felony record from a liability into a résumé-builder at a school known for embracing unconventional educators. He taught a seminar called “Crime and Punishment” and quickly become a fixture at the school.