So I'm living and going to school in NYC. I live about a ten minute walk from campus and I carry my school supplies in a messenger bag. I also carry a few everyday use and emergency supplies:
Thermos
Clifbars
Washcloth
Sunglasses
First aid kit (trauma bandage, nitrile gloves, bandaids, aspirin, Benadryl)
LED headlamp and extra batteries
Spoon and chopsticks
Reusable shopping bag
Change
Notebook and pens
Swiss Army knife in my pocket

Suggestions?

Be mindful I don't want to break my back carrying it, I'm a student without many funds, but I have a decent inventory of camping supplies.

if you don't already EDC...
whistle
bandana
I'd add a small AAA LED keychain light in addition

bag
multi tool or silcock wrench to access water... my Swiss Tool will cut chain link fence
N95 mask
chemical splash goggles/swim goggles
syringe for eye wash or wound irrigation
leather gloves

A Metrocard with a few rides on it.

Are you looking for emergency supplies, or everyday comfort items?

Thermos? Water bottles (full) instead.
Washcloth? Large bandana instead.
Spoon and Chopsticks? ??? What for?
Reusable shopping bag? ??? I guess OK if you're planning on shopping.

LesSnyder already mentioned a flashlight and whistle and mask.

Cellphone (even a retired one without a service plan will work for 911 calls)
Toilet paper (don't you just hate it when the stall is without?)
Fingernail clipper with file (I'm surprised how often I find this useful)
Anything for self defense is probably out for NYC, so no need to mention

It's not true that you can't have anything for self defense in NYC. That's just an exaggerated impression of people who haven't spent a substantial amount of time living in New York. However, this is something I wouldn't recommend any New Yorkers to discuss online publicly. If you get into a lawsuit, you definitely don't want put evidence of your self-defense planning and training in public.

Great suggestions so far.

One thing I see missing is shelter. Even with the genetosity of spirit that abounds in NY, Id be concerned with getting stuck outside for a while because of "X" (fire, bomb scare, etc.) within your local AO. It's Spring, so cold and wet weather is part of the seasonal forecast. I'd add a disposable poncho or something equivalant. Personally, i'd also throw in a winter hat, mitts/gloves and a pair of socks, unless you know you're going to be dressed for the worst weather. (I find that harder in Spring and Fall because the temp variances are much wider.)

Great answers. This is just what I carry, I haven't distinguished between emergency and comfort items. I don't really do so in practice, somethings are more obvious, but everything else is a source of improvisation materials. Even with first aid since aspirin and Benadryl can be life saving. There is a whistle attached to my key chain, and I carry a cellphone, earplugs, wallet with a metro card, ID and an extra $20, and I usually have a (not necessarily clean) handkerchief. I've started checking the weather before I leave and adding a sweater or rain jacket if the forecast looks like I need it. Usually, I'm pretty well dressed anyway though.

N95 mask and gloves will probably be added.

Justification for a few other items.
Spoon and chopsticks, definitely comfort item, but I use them all the time. I also carry headphones...I really don't want to destroy them, but they could be most useful as cordage.

Thermos, I carry this instead of a water bottle since I usually leave the house with coffee in the morning. Then I rinse it out and fill it with cold water. If I'm going downtown or something on the weekends, I'll fill it with hot tea or cold water or lemonade. It's a compromise, but it keeps me carrying it.

Shopping bag, mostly for, well, shopping. When I drove everywhere I couldn't remember carrying reusable bags for anything, now its right with me. But it also gives me the option of picking up supplies if I have the opportunity and carrying them more easily.

Washcloth, I like because its more absorbent. I go back and forth between this and a small towel. Why should I carry a bandana instead?

a bandana is typically large enough to improvise a sling for an arm or collarbone injury, and long enough to wrap around a leg for an improvised tourniquet...and if color neutral a "do rag" if you don't have to worry about color associated street gangs...for Sun protection and holding a bandage on a scalp cut

I EDC a bandanna regularly. A good all cotton bandanna makes a very effective towel. I regularly used mine when showering at work after commuting by bike. A cotton bandanna is one of the few cotton items I still employ, and they are marvelously versatile.