for NY:
does amount of lost income matter, or only that there was lost income?
and, to use unemployment as proof: case was opened, 'waiting week' was registered, but 'back weeks' haven't yet been certified. is there anything there that can be used for proof?
Backrent is only for missed payments.
If you're still tight you can apply for relief going forward.
rent stubs showing a 'rolling' backlog of about a month each month: would be eligible for one (most recent) month's backlog?