It seems like the VSS writers on the Hyper-V host server are not stable. Please check the same by running the command VSSADMIN LIST WRITERS in the command prompt on the Hyper-V host server.
If any of the writers are not stable then please reboot the server and then run the backups.
On the server giving you this issue, open a command prompt and type: vssadmin list writers.
Any writers not listed as Running are in error...normally a server restart fixes this by resetting the writers.
The issue is on the Hyper-V host server. So you need to run this command on the Hyper-V host server.
And instead of restarting one VM you need to restart the Hyper-V host server.
Just restarting the VM will not have any effect on the writers of the host server.
we can clearly observe in both the screenshots that the Hyper-V writers are in Failed state. This is clearly issue with the OS / Hyper-V.
I would suggest that you install all the Microsoft updates that are available for the OS version. If the writers are failing even after that then its better to speak to Microsoft Support over this.
Looks like space issues. try to extend the disk volume that needed to create VSS shadow copies.
How to extend a data volume in Windows Server 2003, in Windows XP, in Windows 2000, and in Windows Server 2008