I don't know if it's possible or not but could I place a made up neighbourhood in a city say New York? Or in any city? Can I change some street names here and there and place shops anywhere I want or does it have to be exactly like how it is on the map? Would people get offended if I didn't get it right? I've only been to NYC once on vacation so I can't really say that I know it very well.
Of course you can! People plunk down fake cities in real states all the time, so why not a neighborhood in a city?
Your last sentence makes me worry just a bit though. This technique works best when you do know the city fairly well. You'll be more successful if you can say where your new part fits in, and if you get the rest of the city right, it will help the reader suspend disbelief.
I've built subway systems and back alleys into cities where they don't exist, but knowing that there's a particular corner eatery on D Street, and bringing that in, will have even native locals thinking, "Is there an alley back there?"
I recommend getting a physical street map (from the time period of your piece), and referring to Google Street View often. Search for things that fit the detail you are looking for, and more importantly, make sure that you don't contradict something important unintentionally. Don't build your neighborhood on top of Wall Street, unless you really mean to.
You're the author. You "can" do anything you like in your work. If you want to insert a made-up neighborhood in New York, you can do it.
But your question is really what the consequences of it will be. Will readers accept it?
Well, anyone who doesn't know New York likely won't notice the change. But people who do know the city (and I understand there are a lot of them) probably will notice.
The real question is this: are you making this change for a reason, or do you simply want to control the background of your work without having to learn about the actual New York?
If this change is a part of the story or plot, then you need to treat it the way historical fiction treats changes to history: bathe everything that you didn't change in as much verisimilitude as you can. Pick a specific district of New York to replace. Then decide how this new one will be different form the old one, and how its neighboring areas will be affected. What class stratification is this district? What are the crime levels? What are the demographics? What is its history, and how did that influence the history of other parts of the city? And so forth.
If however you simply want to set your work in "New York" without doing research on what that actually means... well, you still can do this. And a lot of people may not notice or care. You can build a fictional neighborhood of the city with whatever background you want.
But New York isn't a generic "big city". It has its own flavor, and each of its districts has their own flavor.
People probably won't get bent out of shape if you get street names wrong or make up buildings or whatever. But if you are trying to write a work about "generic big city" and name it "New York", it will impact the verisimilitude of your story to some.
At the very least, you should make it clear to the reader that your insertion of this location is not a mistake on your part. The best way to do that is by putting in real details of New York city. If the surrounding area is legitimate, then it's clear that the different areas are a deliberate choice rather than something you just did.
If it's a "magic" hidden neighbourhood (like, let's say, Harry Potter's Diagon Alley) or something like that, you have total freedom to do what you want with it. If it's not, I suggest you to create a suburb (or a suburban town, they can be relatively big too) instead of a neighbourhood, 'burbs are in the outskirts and more or less far from downtown, so more creative freedom for you. If the real city is by the sea or a big lake/river, creating a small peninsula (or even an island, linked to mainland by a bridge) not far from the city core wouldn't be a bad idea IMHO. Be vague with street names in any case. Then, just write down a disclaimer saying that the X hood/suburb is your personal creation and yadda yadda, so no one will complain.