All Sorts Of Rebooting Problems

This morning my computer rebooted for no apparent reason.
Then a screen said "Please wait" without the harddrive activity light doing anything for a half hour.
I pushed the power button and the computer went off. I pushed it again and nothing happened, repeatedly.
I unplugged it and tried again. Nothing happened. I kept pushing the power button and eventurally it turned on and said "working on update 100% complete", with the activity light on solid.
I came back in 10 minutes and it said "working on update on update 30% complete". It gradually went to 100%, rebooted and said "couldn't complete update".
After it came up, and checked my updateds. The machine says I am up to date, with the date of the last update 2 days ago.

So,
1) Why did it just hang on "please wait"?
2) Why wouldn't it turn back on?
3) What was it doing when my last update was 2 days ago and I am now up to date?

2 days ago it did 3 updates, most prominently '2017-08 Cumulative Update for Windows 10 Version 1703 for x64-based Systems (KB4034674)'

 

It has gotten worse. Got back from walking my dog and found it off. Power button does nothing.
Any advice?

 

Is it a laptop or desktop?
It is a strange request but I would disconnect the battery AND the power source. Click the turn on button a couple of times. Reconnect the power source and press the turn on button. This process has worked for me more times than I care to recall.
Good luck,
s.

 

It is a desktop. I connected the HD to another computer and it looks fine.
Unfortunately the connectors on the PSU and the power switch are different between the two computers so I can't test anything.
Before I buy a new power supply, is there any way to tell which one is bad.

These are two pictures of the PSU. What do I need to buy to replace it. It is a Lenova H050.

 

I googled a bit and found the Lenovo PSU is 14 pin and standard is 24 pin, which is why my second computer is different than my first. Amazon has a $6 adaptor; some people say it works just fine; others say it doesn't give the right voltage. What to do????