When you say that the apartment is "under HPD," do you mean that it is now owned by the City of New York? If so, HPD has its own rules which differ somewhat (but not much) from the normal tenant succession rules under rent stabilization. If you lived with your mother in her apartment for at least two years immediately prior to her death, you should be entitled to succession rights. You may have a problem if you lived there, had an income, and your mother failed to report it.
Most rental agreements with HPD do allow the City to move you to an appropriate sized apartment, though they rarely actually do so. (These are the same people who will take six months to fix a furnace and a year to replace a roof that's leaking. Moving you is not likely very high on anyone's "to do" list.)
HPD is permitted to base your rent on your household income.
If your income is low, I suggest that you try to see a legal aid or legal services attorney. I practiced in that area for 20+ years but the regulations governing city-owned housing are always changing, and legal aid and legal services will be the best people to consult regarding the current rules.
My other suggestion is a practical, not a legal solution: contact your City Council Member. She or he may be able to assist you in going through the city bureaucracy.