Windows 10 doesn't shut down or update

I have had a frustrating and a little bit scary problem with my Windows 10 pc for almost a month now. When I try to turn off the computer, it starts closing and the screens go black. But immediately after they start again and the lockscreen comes up. The only way to shut the pc down is force shutdown, by holding the power button for a few seconds. I'm scared for my hard drives every time. As far as I can tell, restarting looks normal, I get the mobo logo and bios stuff, and everything is normal. On shutdown I don't get the motherboard or windows stuff, just goes straight to lockscreen. Fans spin the whole time.

When I started investigating the problem I noticed that the computer can't install some updates. Windows update gives an error code 0x800f0922. It's stuck on Windows 10 version 20H2. When I boot the pc, it tries to install, but after a restart it says "We couldn’t complete the updates. Undoing changes. Don’t turn off your computer." Then it restarts a couple of times and boots. I don't know if it is because the shutdown issue or vice versa, or completely unrelated.

I have tried everything I could think of and a lot of stuff from the internet, but can't seem to find the issue. All help is greatly appreciated.

I have a Gigabyte GA-Z97P-D3 mobo, an Intel i5 4430 CPU and 64-bit Win 10 installed on a 125GB SSD (not full, 11GB left). I can provide further specs if needed.

Thank you, Eemil