How to "super delete" Mail messages in Mail?

There may be a better way but I would try the following.

Method:

1. Quit Mail on both Macs
2. Locate Mail folder on second Mac (~/Library/Mail) and dupicate it (cmd d)
3. Copy Mail folder from first Mac to second
4. Repeat items 2 & 3 for Mail preferences (~/Library/Preferences/com.apple.mail.plist)
5. Start Mail on second Mac to see if it has worked

Warning:

This should make the second Mac's email exactly the same as the first. If the second Mac had extra accounts etc they will no longer be available until you reverse this. It should be easy to delete unwanted accounts.

If this fails to work it should not be too difficult to reverse everything.

A better way perhaps:

If you use POP you may wish to see my post at: https://discussions.apple.com/thread/5880361?answerId=24804362022#24804362022

POP or IMAP?

I rarely use IMAP but you may find that emptying the trash will remove them from the IMAP server. POP accounts don't normally.

Try selecting this from the top menu:

Mailbox > Erase Deleted Messages > In All Accounts

How are you emptying the trash folder?

I am surprised you arrange to get so much mail you don't want.

If you only want to delete the messages I doubt that it matters how you do it.

I can see no point in having both POP and IMAP accounts.

That option would stop the eMail from loading on another device or computer once you get it the first time.

I don't see any reason for not using entirely POP or IMAP. There is no reason to use both.

Here is an entirely POP solution: https://discussions.apple.com/thread/5880361?answerId=24804362022#24804362022

Are the POP & IMAP for the same account?

OK, not quite clear here yet though, are you trying to delete them from the trash on the Mac, or from the server?