A Hyper-V VM stuck in the Running state with the Backing up status is one of the most common situations that an admin could face. In this situation, Hyper-V Manager will not allow you to start or stop virtual machines.
Rebooting the Hyper-V host typically fixes the problem, although this isn't ideal since it involves VM downtime.
In many cases this may be caused by a problem with the "Microsoft Hyper-V VSS Writer" service on the Hyper-V host. We can easily check this by running "vssadmin list writers" from an elevated command prompt which should return no errors shown in the output, as seen below:
Hyper-V VMMS service and the Hyper-V VSS Writer services are very dependent on each other and that's why it's important the Hyper-V VSS writer should be in a stable state.
If the Hyper-V VSS Writer service is in a stable state and you are still seeing some VMs stuck in a "backing up state", restarting the Hyper-V VMMS service can solve the issue.
You can restart the Hyper-V VMMS service by navigating to the Services in Computer Management or simply open run and type "services.msc", right-click the service "Hyper-V Virtual Machine Management" and select restart.
Once the Hyper-V VMMS is restarted, the virtual machines might disappear from Hyper-V Manager for a short period, however, once the service is up and running again the list with all of your VMs should re-appear, and the backing up status of the troubled VM should be cleared as well.