On the lease, add the housing authority (or any program funder) as a payer with its monthly share, and set the tenant's share. From then on each cycle posts two charges, one for each payer.
Why split them
When a tenant portion is late you want to chase the tenant, not the housing authority, and the reverse. Splitting the charge means payments apply to the right side and the rent roll shows who is behind.
When the split changes
Housing authorities adjust the tenant portion after a recertification. Edit the payer amounts on the lease and the next cycle uses the new numbers. Past charges stay as they were.