Since the name of the recipe if from the town of Hamburg, I guess it makes it a hamburger with ham

Less. A hamburger is a descendent of the Hamburg steak which is a ground beef patty. It's got nothing to do with ham.

Then you just have a ham sandwich.

Probably less. The 'ham' in hamburger doesn't refer to the type of meat. But rather, the 'hamburg' in hamburger refers to the indirect origin of the creation (via the Hamburg steak sandwich). Hamburg in Germany. So putting actual ham in a hamburger would a) make it not really a hamburger, and b) be pretty confusing to people.

Edit: corrected my previous explanation based on the feedback of others.

Less of a hamburger, more of a ham burger.

No, that’s just a ham and cheese.

Steamed hams

What if it's ground ham?

Spamburger

I think I once ate a ham hamburger in a berg near Hamburg.

Well, there's no sand in a sandwich, Manwich isn't a Sweeny Todd product, but a veggiewich is full of veggies. Maybe if you grind the ham and mix a bit of egg and breadcrumb, you could make a hamhambuger, which isn't more or less, just different.

It's a Newfie-burger.

Just be British and call them beefburgers because that is what they are? Except when visiting McDonald's then they are hamburgers..

Haha nice

Please don't advocate murder