Hi EasyAnswersNow,
It sounds like your Google email keeps reappearing on your Mac. We know how important this is for you, and we will help get you to the right place to find a resolution. The first thing is to try the steps below. These simple steps can resolve issues like this.
Permanently delete emails
In the Mail app
on your Mac, do one of the following:
• Choose Mailbox > Erase Deleted Items, then choose an account.
• In the Mail sidebar, Control-click a Trash mailbox, then choose Erase Deleted Items.
You can set the “Erase deleted messages” option in Mail settings to have Mail permanently erase deleted messages for you at a specified time. See Change Mailbox Behaviors settings.
This info is located in the Mail User Guide here: Delete emails in Mail on Mac - Apple Support
All the best.