In Warwick, some worried that the sunflower butter and jelly sandwich could become a marker of shame.

In the days after the district announced the policy, nearly 600 comments have been posted on its Facebook page. Some readers blamed parents for not prioritizing their household expenses. Others said they had received mailed warnings that they owed just a few cents, because a child had put a carton of milk or another item not covered by some meal plans on his or her tray.

“I know as parents this is our responsibility but why take it out on kids if parents are struggling,” one woman wrote on the district page.

“We need to look at both sides of this!” another wrote. “While it is inappropriate to embarrass a child over their parent’s failings — there’s also the argument that if we provide free lunches with no accountability, many parents will purposely choose not to bother paying at all.”

The district also defended itself against reports that it had refused donations from people outside the school.

Angelica Penta, an owner of Gel’s Kitchen, said she had set up donation jars at her restaurant last year, after she saw a report about a girl who was refused food that she had chosen because her account was delinquent.

When Ms. Penta tried to bring a $4,000 check to the school district this year as a donation, she said, officials there refused it. “At the end of the day, it is the child who is suffering,” she said in an interview.