Hi there GeneDeJ,
Thanks for the post in Apple Support Communities. It's definitely important that your Mail app works as expected, and we can help. It looks like your deleted messages are not appearing in the Trash folder.
To verify, are you missing the Trash folder entirely, or do you have the Trash folder but find that deleted messages do not appear there?
It sounds like you may want to start by checking Mail > Preferences > Accounts. Then, select the account in question from the left-hand column and click the Mailbox Behaviors tab on the right. Then, click the "Trash Mailbox" drop-down. If it's set to "None," try changing it to "Deleted Messages." This link has details for reference on how to Change Mailbox Behaviors preferences in Mail on Mac.
Let us know the results.
Take care.