I see that even after deleting an email from the webmail site, the emails deleted repopulate on your iPhone. I'm happy to help.
Restart your iPhone, iPad, or iPod touch -- Restart your iPhone. Once it's returned to the Home screen, test the issue.
Add an email account to your iPhone, iPad, or iPod touch -- If restarting your iPhone didn't resolve the issue, tap Settings > Passwords & Accounts, then select the email accounts you're having an issue with, scroll to the bottom, then tap "Delete Account". Once the accounts are deleted, restart your iPhone. When your iPhone returns to the Home screen, return to Settings > Passwords & Accounts and set them up again.
Let me know how that goes, and take care!