I rent a home in housing in our town. I received some papers that I did not understand (2 days before a court date) and I missed the court date. Therefore the writ of execution for possession was granted to Housing. I am trying to pay them and owe them less than one month behind. They want me out, but I feel it is personal. I know other tenants are allowed to be late. I have only been late. There have been no other issues with my tenancy. But they always got paid.

How much time do I have? I am a single mother with two children. Is there anything I can file at the court to have more time? I received copy of the approved judgement today and housing has not yet told me how much time I have, but I fear it may only be a couple of weeks.

If a court has already issued a writ of execution on an eviction, you have until the time limit listed in the writ. If you don't vacate within that time, they will hire a sheriff to physically remove you. That might buy you a few extra days depending on the sheriff's availability.

Your only other option is to ask the court to vacate the writ because you did not have time to prepare and missed your hearing. Then ask for a new hearing.

... I received some papers that I did not understand (2 days before a court date) and I missed the court date...

I suspect that, if you ask to vacate the writ, the judge is not likey to believe you missed the hearing because you couldn't understand when it was. Most such notices are quite clear on that point.