I have a similar problem to one posted. After I run CCleaner and open Outlook, both of my Gmail accounts require me to sign in again. That would not be so bad but they need to be signed in 2-4 times for each account. This ONLY happens when I have run CCleaner.
I also have Glary Utilities and this does not cause interruptions. If I don't get help, I will drop my subscription to CCleaner at the end of the license.
You could try...
Run CCleaner, leave open after running
Open Outlook and sign in to your Gmail accounts
Go back to CC, open Options/Cookies
Move any cookies associated with Outlook or Gmail to the Cookies to Keep section
For Gmail on its own I keep accounts.google.com, contacts.google.com, google.com, and mail.google.com - whether I need all of these is debatable but its no great overhead.
To be clear
Outlook (in your original post) refers to the free email software that comes with windows, and not the same name Office program or the email provider @outlook.com. Is this assuption correct?
If yes continue.
Within the outlook program you have two gmail accounts. Am i correct in this assumption?
If yes continue
After running ccleaner (health check or custom please tell us) those two accounts require password again to check mail.
Correct? If yes then we can start to weedle down what the cause entry is.