I found a few moving companies in NYC that offer HRA (Human Resources Administration) moving services. Here are some options:

Available Movers: Available Movers is a leading moving company that serves as public assistance movers for customers. These customers qualify for HRA moving assistance. They offer free services to individuals who cannot afford moving and storage costs. Those who qualify for free moving services through state programs can benefit. Available Movers provide emergency services for those facing eviction in NYC as well.
www.availablemovers.com, call: 718-292-6500 They offer movers, drivers, trucks, and secured storage units for qualifying individuals and households.
Site: , call: 201-490-1165

Crossroads Moving: Crossroads Moving is known as one of the best HRA-approved moving companies in NYC. They provide free services to individuals who qualify for HRA moving assistance. Those who cannot afford the cost of moving and storage are supported. Crossroads Moving offer professional assistance with drivers, movers, and trucks. This eases the burden of relocation for eligible individuals.
Crossroads Moving, call: 201-567-4135

These companies offer assistance to individuals who qualify for HRA moving services in NYC. Contact them directly to inquire about their services and eligibility requirements for HRA moving assistance.