There are a couple of options depending what you want to do. Generally Speaking Email is designed to work seamlessly with your mail server account. Therefore if you delete emails in your regular mail client, or in Speaking Email, they will delete in both places.
To delete old mail from Speaking Email and your mailbox on your mail server account, tap Trash. You can say “trash” or “delete“ when Speaking Email is listening for voice commands (the buttons are on screen and the voice command button is green).
Or you can tap Archive (moves into a folder "archive" on your mail server).
To delete it from Speaking Email only, and not your mail server, go into Account Settings, and change the "Archive" command to not do anything (set the switches to off). Then use "Archive" command to remove them in speaking email without affecting your mail server. There is a shortcut double-tap to archive.