And that's a picture of an already existing go-kart you own, right?
As far as putting power down, there's any number of methods I know of, and I'll link you some right below here with the name of it an how it works.
A basic overview of how the locked and open differentials work and came to be. This will only work if you put the differential between the rear wheels and split your axle, in order for the gears inside to do their stuff. I add that last part because the picture in OP gives me reason to believe your wheelbase is too narrow for one of these.
In the case of that, what seems to me would fit best is a direct drive system. Mounting your engine/motor sideways so the crankshaft is pointing straight east or west of the car compass, and having a gear on it connected to another gear on the axle via a chain, much like a bicycle. If your motor has a throttle response to it (if you can control the input of fuel to control the output of power) then this would be perfectly fine, as well as if you had an electric motor which acted similarly based on the amount of power flowing through it. For the sake of simplicity, I WOULD suggest the direct drive system. Those disc brakes on it though make me hesitate to suggest such. Is the other side also housing disc brakes, or just the one side?
