Wrong oil in my lawn mower
Refill it, the trace amounts won't be an issue. Hopefully it should stop smoking after a couple of minutes running with the correct oil. If not you can shut it off and look at other causes.

It may be worth your time to inspect the muffler and replace it if needed. They do corrode over time which leads to noise along with the possibility of trapped combustion byproducts which can smoke or be smelly. If you've got one of those torpedo style ones it's also a plus if you can replace it before it crumbles and becomes much more difficult to unscrew from the engine.

Checking out the spark plug won't hurt either, since a fouled plug doesn't help combustion. It's not a big cause of smoke, but it's a real minimal amount of work to check.