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No, Section 8 Housing Choice Voucher tenants do not need to report the income they get from the government stimulus check to their housing authority. Under the Housing Choice Voucher code of federal regulations, these types of payments are not considered as part of a participant’s annual income.
According to HUD’s FAQ’s for Public Housing Agencies (which is geared toward answering questions for housing authority management staff), stimulus payments made directly to individuals and families would be excluded from income, as they are temporary, non-recurring payments (per 24 CFR 5.609). HUD’s entire FAQ for Public Housing Agencies can be found here.
For information on similar questions, read: Do unemployment benefits affect Section 8 and Public Housing income requirements during coronavirus?