How can I fix my nonprofit mission?

I think there simply may be some confusion over the question you asked.

If you are in the USA, the hospital is required to have a "charity care" component that you can donate to. This is a bucket of money that the hospital will distribute to the patients/their family when payment help is needed. When you donate money to this fund you can't specify who the money goes to - like you can't say this money is to be used for Shreyaas' with their medical bills. However, as a non-profit, you cannot give money to another organization to be used for a specific person either.

As a non-profit, you would be required to generally college money (not collect money for a specific person). If you want to help Shreyaas' with their medical bills, you wouldn't "donate" to the hospital. You would pay the bill on Shreyaas' behalf.

Do you see what I mean?

Do you have income for your non-profit business? If you don't, then you don't have a business. And the government is not going to award you a non-profit status without completing all off the numerous legal requirements involved. Since it seems that you don't have a clue about how to start a non-profit business, I suggest you get a job and designate 25 percent of your annual income to charities. Since people have the option of obtaining insurance for their medical and what medical isn't covered and is unaffordable is covered under state government plans, consider giving to animal causes, or at least some of it.

Donate to the fund that supports cars, utility, food, etc.,
Most hospitals, even public ones, are for profit. Why would you donate to a for profit business?

you can't.
You must be at least 18 to create a nonprofit corporation.
You have to file form 1023 with the IRS and jump through a hundred other hoops.
read this.
https://www.irs.gov/charities-non-profits/life-cycle-of-an-exempt-organization

Are you asking for help to create a mission or trying to decipher how to add the other causes within the mission statement? Also, is this the only hospital you can donate to? Perhaps there is another that will allow patients bills to be paid anonymously. Perhaps HIPAA laws prevent the facility from being able to accept the funds...not sure???