How to permanently delete Gmail or Google Apps email from Mac Mail
You should have already enabled IMAP in Gmail via Mail Settings > Forwarding and POP/IMAP > IMAP Access. Auto-Expunge should be ON by default. Don't forget to "Save Changes" if you've adjusted anything.
In Apple Mail, under Mail > Preferences > Accounts > Mailbox Behaviors, make sure that both options are checked "On" for Trash. (Move deleted messages to the Trash mailbox; and Store deleted messages on the server.) The option to "Permanently erase deleted messages when:" can be set to whatever duration you wish, but I'm using "One day old" to give me a few hours to change my mind if I need to fetch a message back from the dead.
Next, in Apple Mail, you will need to "Show" the sidebar that lists accounts and folders if it's not visible already. Find the Gmail account, and look in the [Gmail] folder. Select the Trash folder, then choose in the top menu Mailbox > Use This Mailbox For > Trash. (While doing that, you might also want to choose to map the other folders Drafts -> Drafts, Sent Mail -> Sent, and Spam -> Junk, although none of those really has anything to do with this issue of permanently deleting mail.)
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