There are two main functions to paddling a surfboard or racing board lying down:
Positioning on the board
Good posture
Every board is a different size so you need to position yourself so that the board lies flat in the water. If you are too far forward, your nose will push under the water. You don’t want this. You will need to see the nose of the board just out of the water.
If you are too far back, the nose of the board will be sticking up and you will not be able to move the board very fast. The board needs to lie flat in the water with a slight tilt back.
Your posture needs to be set up, by lifting your chest off the board and engaging your core muscle groups in your back and stomach. If your posture drops you will lose your balance.
Your arms and hands need to be relaxed as you paddle like a freestyle swimming action.
Make your strokes long and strong.