The speed of sound of 340 m/s is measured with respect to air. So here we can mention two cases:
If you are inside a car that is moving at 340 m/s in the highway and the windows are closed (so that the air inside the car is also moving at that speed), the sound of your clap will move at 680 mm/s with respect to the ground. This is because it moves at 340 m/s with respect to the air inside the car.
Imagine the same situation but with a car that has no roof nor windows. Then you, inside the car, will find that the air moves at 340 m/s backwards. If you clap in this situation the sound waves will move at 340 m/s in the air, but because this air moves at 340 m/s backwards you will see that the sound waves do not move at all.
The second case is exactly what happens in this picture:
When the supersonic aircraft is moving exactly at (or very near) the speed of sound (both with respect to the air), the waves travel at the same speed and they accumulate to form that peculiar cone.
In the case of light the case is different because light speed is 300000 km/s with respect to space itself. And you cannot move with respect to space, you move in space with respect to other things. Motion is relative. So the speed of light is the same for everyone independent of the speed of the observer (or the source).