It sounds like your mower has been re-engined at some point. Does this red lever kill the engine if moved fulley forward? If so, it's not supposed to be a throttle, it's simply an off switch. There should be a spring link and a rigid rod attached to the governer arm. The spring should be attached to a fixed point at the front of the carb, and the rigid link should go to the carb throttle. This type of engine is (was now, since Tecumseh are defunct) fitted to pressure washers (among other things), and it's fixed speed.
All of that, however, doesn't explain why it cuts out. Check that the fuel tank cap vent isn't blocked, allowing a vacuum to form in the tank, then check there is good fuel flow to the carb (pipe can disintegrate internally, but look OK on the outside). After that, you need to check if you still have a spark immediately after the engine dies. If not, you have a coil that fails when hot (not unheard of). If you do have a spark, the only answer is a thorough carb clean, or replacement.
phil_saunders(a)bigfoot.com
Phil Saunders, June 2011
