Overpaid by almost 7k (mod) England

Hi, I'm in the forces and in my monthly pay (dec 25) I got overpaid by almost £7,000 due to an unrequested MDR salary advance. I understand this is most likely an accounting error, I have email the relevant department to highlight the error and I have set aside the money to repay it.

My question is however what are my rights in regard to getting any money back which I have lost due to their error. This advance pushed my monthly salary over 10k which has resulted in me paying around 730ish in tax compared to the normal 400ish, also although minimal I've still payed an extra £50 to student loan repayments both of which I wouldn't have incurred if it wasnt for this error.

This amount may seem small to some but im the sole provider for my family including two children under 2. So if they take back the overpayment but im still left out of pocket by 300-350 im going to struggle this month.

This has also affected my wife as well shes on unpaid maternity leave and gets a small amount of universal credit maybe 300 which she uses towards the children. This amount is based of duel income so she won't receive anything next month due to the error inflating my wage.

Any advice would be welcome