If you’re repeatedly prompted to enter your username and password for the Gmail account in Outlook, it’s possible that the login credentials you’ve entered are incorrect, or the authentication settings in Outlook need to be adjusted. Here are a few steps you can take to troubleshoot this issue:

1. Verify your login credentials: Double-check that you’re using the correct username and password for your Gmail account. Make sure that you’ve entered your email address and password correctly, and that you’re not accidentally including any spaces or special characters.
2. Check your account settings: In Outlook, go to File > Account Settings > Account Settings, select your Gmail account, and click “Change.” Then, go to the “More Settings” > “Outgoing Server” tab, and check the box that says “My outgoing server (SMTP) requires authentication.” Make sure that “Use same settings as my incoming mail server” is selected.
3. Enable Two-Factor Authentication: If you’ve enabled two-factor authentication on your Gmail account, you may need to generate an app password to use in Outlook. To do this, go to your Google Account security settings and create an app password. Then, use that password to log in to your Gmail account in Outlook.
4. Disable Antivirus or Firewall software: It’s possible that your antivirus or firewall software is blocking Outlook’s access to your Gmail account. Try temporarily disabling your antivirus or firewall software and see if the problem persists.
5. Clear Outlook Credential Manager: In the Windows Control Panel, navigate to “User Accounts” > “Credential Manager”. Look for any entries related to your Gmail account and remove them. Then try re-adding the account in Outlook.

Hopefully, one of these steps will help resolve the issue with repeated prompts to enter your username and password in Outlook.