Lump sum Separation pay received 2 months after Michigan unemployment claim filed.

My employment ended mid October. I was told by payroll to file for unemployment right away, but waited to file the following week since I received my final pay Friday of the same week employment ended. A separation agreement was signed by me and the employer late November; 4 and 5 weeks after employment ended. I received an email 2 months after becoming unemployed saying the small lump sum payment was being sent. The separation agreement states the lump sum payment is in no way severance pay. However, I believe Michigan may still view it as such.

Since it was a lump sum payment, processed via their payroll with taxes withheld, am I required to payback the unemployment received prior to the agreement being signed and lump sum received? It was not allocated for any specific weeks.
How should I report it in unemployment system since I don't intend to file a claim for the week I received it? I'm sure since payment was processed by payroll and taxes withheld, the state will have record of it as income.

Summary:
Unemployed October
Separation agreement signed late November -* states not severance pay
Lump sum received mid December * 2 months later/no allocation