HMRC clearly did check the tax return as you are being asked to repay the excess tax repayment, together with a penalty. A notification of the repayment would have been sent to you when it was originally made, though you appear to have not received that and the money has gone. The tax repayment can only have occurred because of a mistake in the return and you should double check that there was a mistake.
I'm afraid that there is no way that HMRC will accept £80 per month to repay the tax since it will take nearly five years to repay at that rate and interest will be charged on top. Unless you can negotiate a payment plan that is more acceptable to HMRC, they will continue to write to you themselves or hand the matter over to a commercial debt collector which you really want to avoid at all costs and eventually they will take you to court. You don't have to allow anybody into your home unless they have a court issued warrant.
If I were you, I'd try to think of a way to raise the money from family, firends or your bank to clear the debt as quickly as possible. Read the notes here and either call the Business Payment Support Service on the number here
or see the local Citizens' Advice Bureau who deal with many tax debt problems. Some local councils also have debt advice departments. Taxaid
may be able to help but they normally only deal with people on very low incomes. I'm surprised that HMRC don't appear to have asked you to complete and income and expenditure form, give that you've told them that you cannot afford to pay more than you have already offered.
I'm afraid there are no easy answers but you should not let the matter fester. As I said earlier, the last thing you want is for a private debt collection firm dealing with it. Sometimes, making a complaint can buy you time but not always. You can read about doing that here
. If you do that, you might cite the fact that HMRC have not asked you to complete an income and expenditure form to back up your offer.
I hope this helps but let me know if you have any further questions.