Here's a pic from the web of the same kart.
It's the simplest of throttle cables, just cut a short section of bicycle brake cable with the inner cable longer than the outer sheath. The cable connects to the Briggs's governor linkage under the gas tank. There should be a small cable clamp under there to retain the cable sheath...
The throttle linkage seems ideal, with most of it made of steel rod which ensures it won't snap or wear or abrade from contact with the trail, rocks or other obstructions, also eliminates some of the stretching along its length.