Hello A-V-C,
Welcome to Apple Support Communities. It sounds like you're having trouble with your email accounts on your iPhone. I'd be happy to share information that may help.
If you're seeing your Gmail in the Mail app on your iPhone, and you don't want your email appearing there, you can remove the Gmail account from the Mail app by navigating to Settings > Passwords & Accounts > Gmail > Delete Account. This will stop any of your Gmail messages from appearing in the Mail app. If you're using the Gmail app on your device, you'll still see the email there. You may find the information here helpful: Use the Mail app on your iPhone, iPad, or iPod touch.
If this does not resolve the issue you're experiencing, please reply with a few more details:
-You mentioned that you're seeing mail in both places. Besides the Mail app, where are the emails appearing?
-Did you make any changes to your iCloud email address prior to this issue occurring?
Take care.