My own knowledge of NYC is rather limited. Perhaps the most memorable tale I can tell is of a gloomy, rainy October night some years ago. This is the way I would describe it. It was a dark and stormy night; the rain fell in torrents — except at occasional intervals, when it was checked by a violent gust of wind which swept up the streets, rattling along the housetops, and fiercely agitating the scanty flame of the lamps that struggled against the darkness.

"When I watch TV or a movie and they show a street that has is not a through street, I can tell in 1/2 second whether it is a set or shot on location by the stores and street lights. And so can probably 85% of NY'ers."

IMdonehere... your are so right.... I've been watching Seinfeld reruns and I can tell immediately when they're on a set. (out in the street)

"The axiom "Write what you know" is highly restrictive and does not take into account one's imagination or the ability to research."

Good research can only enhance one's creative abilities. Asking a bunch of folks on the internet is not "good research." A broad question now and again is okay but the OP has asked very specific questions.

"Where do kids from low income families go for good education? Which borough etc."
And "What train do they take..."

Sure someone might know but how does the OP verify this knowledge? Is it important that s/he does?
Well, if they want to impress a New Yorker it may matter.

So maybe, "write what you know" is limiting. "Write what you have researched and verified"

The OP would be well-advised to use the tricks of the better known historical fiction writers as there are some similarities.

Among the most important things is to integrate the research to create a natural sense of time and place so they become their own believable characters. It is easy to say and extremely difficult to do well. And make sure the prose did not scream, "See I did my research." The action, limited descriptions, and events should all say that for the writer.

If you want to write about New York, go back, stay with your family and explore some of things you are interested in writing about. Then, write from the perspective of being a newcomer to the city, who doesn't know it. That would be more believable, and possibly very interesting.

Looks like a hit and run poster... Have fun writing!