(Former New York City public school teacher here.)
Farina is right about hot lunches, but that doesn't necessarily make her right about school closures. In many low income neighborhoods, public schools provide the important social service of providing hot meals to students who would not otherwise have access.
That said, providing hot food is not inextricably linked to a full school day. New York City public schools also offer breakfast and lunch during the summer as well.
Part of what makes this a hard problem to solve is that the city's school system is made up of a diverse group of adults and children. On one hand, you have the parents who can't take off from work because they risk losing their job. On the other hand, you have special needs students that are bussed around the the city with their safety put at risk.
Many—but certainly not all—students attend school in their neighborhoods. My wife used to work at a school that was literally in the center of the housing project where the majority of the students lived. In this case, there is no reason they couldn't make it to school.
But to keep a school open, you need teachers. Many teachers do not live in the communities that they teach in. Some don't live in within city limits. Hoboken and Jersey City (two neighboring cities that are closer to Manhattan than most of Brooklyn and Queens) were closed. What do those teachers do with their children? I've worked at schools with teachers commuting from as far as central New Jersey to get to work every day.
A small subset of teachers will call out, but there is a lot of social pressure from the school administration not to do this. This is mostly because snowy days where schools are open tend to be absolute pandemonium.
To be honest, little to know learning happens on these days. It's a mess of combined classes, substitute teachers, modified schedules, and usually a movie in the auditorium as administrators scramble to figure out how bussing is going to work at dismissal.
There needs to be some options between fully-operational and fully-closed that better serves the needs of students and teachers based on the conditions outside. Hot lunches for nearby students in need does not necessarily mean that we need to buses full of students traveling across the city on icy streets.
Part of the solution is for parents to keep their children home. I'm not sure I understand the vitriol Farina and De Blasio received from parents. Just because schools are open, doesn't mean that you have to send your kids. If you feel that it's dangerous out, keep them home. As I said, little to no learning is going to happen anyway. If you have to go to work, schools are open. I think the bigger issue of school closures is how it affects teachers, custodians, administrators, school safety, kitchen staff, etc.