I live in NY.

I live in a subsidized apt with my 2 young kids and there has been a roach problem now for a few years and nothing has been done about it. I have let the agency know about this numerous times and asked for permission to move and nothing has been done. I recently called the Dept Of Health for them to step in and now I was notified that there will be an exterminator comming in next week.
I have finaaly been approved to move because there is a situation where I have to be closer to my family to help with my kids and I owe the agency almost $4,000 in back rent. I honestly dont think I should have to pay all of it because they have kept me and my kids here knowing there was a problem, am I right? They told me I need to pay all the back by Sept. or I will have to renew the lease again for another year. Is there some way I can get out of this?