-- if not, if I just delete the iPhone mail account, will the missed emails show up in Mac Mail on the home desktop.
It's possible your iPhone client is doing something with the emails, marking them on the server in a manner that the Mac Mail client is hiding them from you.
Are they still visible on the iPhone, even when you can't see them on the Mac? Do you see them in Web Mail?
I'm not a Mac user, but I'd check to see if there is a way to see all mail that it sees, regardless of which folder. For example, I use Thunderbird, and I can search by various criteria, across folders.
Deleting the account from the iPhone might help diagnose this--if the iPhone is not retrieving emails at all and they start showing on the desktop, then likely the iPhone is doing something with them.
Also, could this be something in iCloud synchronization between the iPhone and the Mac, where the Mac may be hiding email that has been viewed or had actions taken? Can you try a different mail client on the Mac? Thunderbird is free and supports IMAP, and doesn't sync with other devices through iCloud.