Will a car with a blown engine turn on?

You'll still have electric power until the battery runs down. As for the engine, it depends on how it is blown. If it has a catastophic failure such as broken piston rods it won't turn over and run. If it is jambed in some way it won't run. Now if you have a cracked piston or a wrist pin about to fail or a bearing about to fail it may run for a short period of time before it completely lets go. If you have a blown head gasket it will start and run for awhile but it will overheat. So in some situations a blowm engine will run, just not for long.