Beth, a few things to separate. School speed camera summonses in NYC are civil, not moving violations—so no points and no insurance impact. The fine is $50, plus possible late fees if you wait too long. You can request a hearing online. There is usually a defense option like 'vehicle was not in school zone' or 'sign was missing/obstructed.'
The video link is timestamped. The problem: the city's position is generally that the school speed limit extends until the 'end school zone' sign, but there is some tolerance in sign placement. If the camera is 30 feet before the end sign, that's still likely inside the zone. What's more likely to work is showing the sign was obscured, turned, or missing. Time sometimes matters too, but many zones are now 24/7, and 7:40 on a weekday is definitely in the active enforcement window.
I'd check the video to see if the school speed limit sign is visible as you pass. If not, go back and photograph the same direction at a similar time and light. Don't assume an admin judge will accept a passenger saying 'the zone ended right after'; photo or video evidence is stronger. If you lose, it's still no points—just a city fee at that point.