I can't add a back up account to my HTC 10. can someone help?

[Android Central Question]

It's been like two weeks. And my phone first started switching off at about 46 percent. Then over a few days it would switch off with a higher charge remaining. I can't switch it back on if I don't connect the charger. And when it does switch on after like a minute, it says 1 percent remaining. My phone starts heating up like an iron all the time. And what's the most irritating is,
1- I get notifications saying "your data isn't backed up". I click on it and I'm asked to add an account and to sign in. After entering the gmail and password I click on agree terms. The screen refreshes and back to home screen and the notification is still there. I cannot add an account no matter how hard I tried. I can't remove account and add again like people say here on the forum because it shows no account at all in the first place.
2 -The second issue is I keep getting a notification saying "sign into your HTC account again". I enter my ID and password, it refreshes and I have to enter the details again.

I have tried to fix 1 and 2 by re entering over a million times but it doesn't work. I can't see my saved videos and stuff on my YouTube account though I can browse. I can browse google and use my Gmail account inbox. Signing in from the laptop to gmail is not an issue and I can't use Google Play Store. My saved contacts have no names now. Please help for the love of god cause I am getting really frustrated..

 

[Rukbat]

The battery is gone. That has to be replaced before you can go any further. (And once you do that, and sign into any accounts you're not still signed into, everything else should work, including backing up.)

If you've been draining the battery until (or almost until) the phone shuts off, stop doing it. Recharge at absolutely no lower than 30%. (40% is even better for longer life.) And don't charge past 70%. (That's just the way lithium batteries are. They weren't chosen because they can be deeply discharged, or because they can be fully charged, just becauise they're light. You don't want a 3 pound battery in a quarter pound phone.)