Yes, because it is not business income and the city improperly put the amount in Box 7, you can likely avoid an IRS notice by reporting the 1099-MISC on Schedule C with an offset of the name amount as an expense (resulting in $0 profit).  I would enter the 'expense' as an 'other' expenses with a description of where the amount is being properly reported.

As for how to properly report the actual loss and payment, it is sounding like it could be payment to offset a Casualty Loss, but you may need to look into the details a bit further for that.