The whole overpayment thing is a little confusing isn't it. It's quite difficult to offer really specific advice because I don't know the full circumstances
You really need to take some action though - just waiting for a breakdown may land you in hot water.
My advice would be to dispute the overpayment, using the COP26 leaflet you have been sent stating all the reasons why you disagree that you should have to pay it back. A few things to remember when you are disputing an overpayment though
1. In a joint award both applicants are 'jointly and severally liable' This means that if there is an overpayment you are both liable for the debt. HMRC don't really care which one pays it back as long as it gets paid back. A bit like defaulting on a joint mortgage or loan, the banks don't care how it's paid or who paid it's paid by as long as it gets paid. So no point stating that as you didn't physically receive the money, or as you spent it on the family anyway you shouldn't have to pay it back
2. If your grounds for dispute are (as I think they may be) that you told HMRC of the household breakdown as soon as it happened but they failed to use that information - hence the overpayments, then you must try to be specific about dates. For example, if you separated 6th April and told them this the same day, but received a payment on 9th April, well that is just unfortunate because it is too late to stop the payment to the bank, however if you didn't tell them until 6th May and another payment was received in May, then the delay letting them know will be what caused the overpayment. Therefore a dispute will be rejected. If you told them 6th April, but they didn't process this until 8th May, then that is good grounds. Be specific with your dates.
3. Another possibility of how this occurred could be that you told them in time, and they finalised your award correctly based on the information held on their systems. Hence the May payment could have been a 'balancing' payment. However it is a requirement that a declaration of circumstances is done at the end of each tax year. If you did not do your renewal/declaration for the previous year, the payments in the new tax year will all be fully recoverable. As you separated in April it sounds most likely that both yourself and your ex didn't do the renewal/declaration for one reason or another. A lot of people think they don't need to in your circumstances because they are not claiming as a couple any more. Unfortunately that means an overpayment dispute is unlikely to succeed if this is what happened.
It's quite a complex area, but in a nutshell, you need to remember that if you think you were overpaid by HMRC , completely their fault, and that it was reasonable to assume that you did not know you were being overpaid when you received the payments, then you have good grounds to dispute paying it back. But be specific with the grounds.
I am surprised though that a debt recovery agency is being used by HMRC - they usually attempt to recover tax credit debt from their debt management department, however you must contact them and tell them you are going to dispute the overpayment, and discuss your options with them. They may be able to suspend overpayment recovery for a few weeks while you complete the form and they investigate it, or it may be that you have to set up some kind of payment plan with them while it is being investigated. Talk to them. You don't want to end up in court for the sake of a phone call
And good luck