This is a very complex process designed to weed out people who don't qualify. Everything has to be done "by the book." If you have tenancy that you can't document and salary you can't document, you're going to have to jump through some hoops.
You will need to take your mother off the application if she won't get an ID card. Here's what the Marketing Handbook says about changing your household composition, and I've added the link below so you can read the whole thing:
2. Changes to Household Composition While Applicant is in Process for a Unit
a. Changes to household composition prior to document submission
are acceptable, provided that the change does not affect the
applicant’s qualification as a household or their ability to meet the
household size requirements for available units within the project.
(See Section 5-3 “Qualification of a Household.”)
b. In the case that an applicant appeals an ineligibility notice citing a
change in household composition, and such change would make
them apparently eligible for an available unit, the Marketing Agent
must send a Document Request Notice, Attachment H-1A.
c. After document submission and after eligibility is determined,
changes to household composition that are not due to an
Extenuating Circumstance will result in the application being
rejected. The Marketing Agent must send them a Rejection Notice,
Attachment H-4.
C. PROOF OF EXTENUATING CIRCUMSTANCES
1. The Marketing Agent will require the applicant to provide, at the time of
appeal, documents to support any changes in household composition. A
notarized statement alone is not sufficient documentation of an extenuating
circumstance, except when permissible in subsections 2 and 3 below.
Common examples of documentation for extenuating circumstances
include but are not limited to: birth or death certificates, hospital records,
divorce records, and court records.
2. Legal or legitimate separation
a. Examples of proof of a legal separation would be either (i) a duly
executed and acknowledged separation agreement filed with the
county clerk where either spouse lives, or (ii) a judgment of
MARKETING HANDBOOK, SECTION 4: OUTLINE OF PROCEDURES
5-4: Income Eligibility
Page 43 of 70
separation granted by the New York Supreme Court, or (iii) proof of
Divorce filing evidencing that a Summons and Complaint and an
Affidavit of Service have been filed, or (iv) legal documentation, such
as tax returns or other government-issued documentation,
establishing one partner’s absence from the household for at least
six months prior to date of eligibility appointment and a notarized
affidavit from the applicant representing that they are separated and
have no intention of reuniting.
b. If legal proof of separation or divorce is not available to the applicant,
they must provide an explanation for the lack of legal proof as well
as documentation supporting such separation or divorce.
https://www1.nyc.gov/assets/hpd/down...ing-handbook.p