Hello there iPhone_Move_Questions,
Thank you for using Apple Support Communities!
I understand that you are having issues with not being able to access the Gmail inbox using the Mail app on your iPhone after restoring from a backup. I am more than happy to help you out with this.
Since you're seeing your other mailboxes, please try to remove the account, and set it back up again:
- On your computer, sign in to your email provider’s website. Make sure that all of your email is there or make sure that your email is saved somewhere other than your iOS device.
- On your iOS device, go to Settings > Passwords & Accounts.
- Tap the email account that you want to remove.
- Tap Delete Account.
- Add your account again."
If you can’t send email on your iPhone, iPad, or iPod touch
Take care.