I often go to KFS, Burger King and MacDonald's and the English name for this sandwich or burger is "hamburger" or "hamburger sandwich".
Wikipedia says: "A hamburger or burger is a sandwich consisting of one or more cooked patties of ground meat, usually beef, placed inside a sliced bread roll or bun. The patty may be pan fried, barbecued, or flame broiled. Hamburgers are often served with cheese, lettuce, tomato, bacon, onion, pickles, or chiles; condiments such as mustard, mayonnaise, ketchup, relish, or "special sauce"; and are frequently placed on sesame seed buns. A hamburger topped with a slice of cheese is called cheeseburger."
Basically of your choices "ham sandwich" and "sandwich with ham" are also possible, however, the latter is ambiguous as @Andrew stated in his answer. "Ham sandwich" would mean that there's ham inside your sandwich and would the best option.
