My deleted emails all reappear

Go System Preferences >> Internet Accounts >>and look on left as listing for " Google " Remove the Google from there.

Add or remove email accounts in Mail on Mac

Use the Mail app to send, receive, and manage email for all of your email accounts in one location. Simply add the accounts—such as iCloud, Exchange, Google, school, work, or other—you want to use in Mail.

Add an email account to use in Mail

• The first time you open the Mail app  on your Mac, it may prompt you to add an account. Select an account type—if you don’t see your type, select Other Mail Account—then enter your account information.
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• If you already added an email account, you can still add more. In Mail, choose Mail > Add Account, select an account type, then enter your account information. Make sure the Mail checkbox is selected for the account.
• If you’re using an account on your Mac with other apps, such as Contacts or Messages, you can also use that account with Mail. In Mail, choose Mail > Accounts to open Internet Accounts preferences, select the account on the left, then select the Mail checkbox on the right.

Temporarily stop using an email account in Mail

1. In the Mail app  on your Mac, choose Mail > Accounts.
2. Select the account, then deselect the Mail checkbox.
3. Now the account’s messages are not shown in Mail.

Start using an email account again in Mail

1. In the Mail app  on your Mac, choose Mail > Accounts.
2. Select the account, then select the Mail checkbox.
3. The account’s messages are shown again in Mail; they don’t need to be downloaded from the server again.

Remove an email account from Mail

When you remove an email account from Mail, the account’s messages are deleted and no longer available on your Mac. Copies of messages remain on the account’s mail server and are still available (from webmail, for example).

Important: If you’re unsure whether messages you want to keep are still on the mail server, move or copy them to a mailbox stored on your Mac (the mailbox appears in the On My Mac section in the Mail sidebar) before you delete the account in Mail.

1. In the Mail app  on your Mac, choose Mail > Preferences, then click Accounts.
2. Select an account, then click the Remove button .
3. Note: If the account is used by other apps on your Mac, you’re asked to remove the account in Internet Accounts System Preferences. Click the button to open Internet Accounts, then deselect the Mail checkbox for the account. To stop using the account in all apps, make sure the account is selected, then click the Remove button .

Sorry but now you have added the Key Part of the missing information from All the previous posting - Remove the Account and Add it back. What do you expect it to do ? Not download the messages on your G-mail account ? If the Apple Mail is not syncing the delated message from G-Mail - suggest contacting Google for the solution as it is their mail service. Apple Mail is only doing what you are asking it to do.

This Apple Support topic talks about Adding an E-mail Account. Just reserve engineer the process to Remove the g-mail account from Apple Mail Application.

I C - from where did you Add the G-mail account and to what Application ??