emails from instagram

Backing up your phone doesn't affect getting emails.

Have you tried deleting the account(s), restarting the device, and then adding the account(s) back?

Mail – Setup on a device    -

Mail - If you can’t send email on your iPhone, iPad, or iPod touch.  *

Mail - If you can't receive email on your iPhone, iPad, or iPod touch    *

Hi there,

Unless you are using an iCloud (@icloud.com) email, your email is not stored by iCloud.

For more information: What does iCloud back up? - Apple Support