This is not what happens. Poor people in NYC tend not to own cars and instead use public transport, walk or cycle. Even if your case is true this is good because:

1. fewer people drive (which is good because less pollution, reduced congestion, fewer pedestrian casualties)

2. as fewer people drive people start to question why there are so many empty streets (and push for that space to be given over to other people)

3. because fewer people drive they are not going to vote for pro-driver policies, like dedicating an insane amount of the city to roads and instead will support things like more walkable neighbourhoods and more cycle paths/lanes. Bikes are a mathematically better way to travel in cities because you can easily get 5-10 times higher capacity!