I can start the mower, but once it is started the key will not stay in the on position, it keeps dying

Check the bottom of the carburetor for a little solenoid. If it has one, the battery may be dead enough that it can't keep the solenoid open causing the carb to shut off the fuel flow causing the unit to die. The easiest way to check if it works is by turning the key to the run position and listen and/or feel for the solenoid to click. If it clicks it is working and there is another problem, if it doesn't then replace it. I recommend having the battery tested or replaced too

If you need further help, I’m available over the phone at https://www.6ya.com/expert/kelly_69a0cf1b13789df1

ya get a new one

Check to see that the starter cable isn't pushed in or out a bit too far and causing the kill switch to engage. It is only held in place by tie wraps and can cause the machine to act as if it can't be charged and/or won't run. Once the cable has been adjusted into the correct position, everything works as expected.

Check the gas cap vent. A partial vacuum in the tank will cause these symptoms.

try replacing the ignition barrel sounds like the ignition swith is goosed.

Will it stay running if you keep the pedal depressed. If so you have a Idle Air Control valve issue. If it wont run at all it may be a bad fuel pump

It could be a needle valve sticking or a mal-adjusted float but more common is a fuel filter that is restricted, once you use up the fuel in the line and float bowl it runs out and dies. A less common but easily tested issue is when the mower has run for the amount of time it begins to sputter and die, loosen the cap on the fuel tank just a bit, if it helps or resolves the issue the vent in the cap may be clogged.

My mazda b2200 starts and will stay running as long as the key is in the start position, but stops as soon as the key is turned from the start position

You are gonna need a New Ignition. Will solve this Problem.

bad ignition switch