Help pay off a sudden debt caused by wage overpayment

Hey there! Back on March 31st, I was let go from my job at a thrift store benefitting a charity after an extended probationary period due to various issues, most of which came down to my untreated undiagnosed ADHD. Normally at that job, I was paid on the first of the following month which accounted for the previous months work. E.g. Work all thru January > Paid 1st Febuary for January's work. However, my leaving on the 31st caused an issue for the admin side of my former employer - somehow - wherein because I was due to be paid for March the following day, I wasn't taken off the payroll immediately. So for the entire month of April, I was on their payroll as normal, and on the 1st of May, I received a payment from them. Personally speaking, because of my poor short term memory and general slow-on-the-uptakeness, I didn't think anything of it; I knew I was due to receive another one that covered the remaining paid time off I had anyway, and I didn't know how much it'd be in total.

A week later, I received an email from the payroll department of my former employer, stating I had been overpaid £580.81 and I had to repay them as soon as possible, either in full or via a payment plan. I've repeatedly begged and asked them for ways around this - either by reducing the amount or by writing it off entirely - but to no avail. Now that I'm unemployed again, it is highly unlikely that I will be reemployed any time soon because I'm on the whole very unemployable for a variety of reasons. Add in the fact that the Job Centre recently declared me temporarily unfit for work for the sake of doing another work capability assessment and that I'm still in the process of appealing the previous decision made from my initial attempt at an ADHD diagnosis, and yeah, it's a mess. The Job Centre will give me unemployment benefit money, but it's generally just under £400 a month and almost all of that is gone just from bills and groceries and other stuff alone; it barely lasts me the full month most of the time. A 12 month payment plan for this would axe an entire £50 off my monthly palty amount of money which is just unfeasible to me. A 24 month plan would take £25, of course, but that means being indebted for 2 entire years which, as you can imagine, is not exactly something I wish to be lumped with.

Currently, they're expecting an email back from me to confirm how exactly I will be paying this back. I haven't responded yet because I don't know how long this is going to take. But, of course, the sooner the better. I really don't need the added stress of this entire debacle, especially when the entire thing is overtly not my fault. And yes, this entire scenario is completely legal under UK law. I checked. Capitalism is hell. Please salvage me from it, at least temporarily. Thank you. I love you.