When working with the Puppet Warp tool, remember to set your Pin Depth. This allows you to control what part if your image shows when two points overlap.
To set the Pin Depth, select any pin while the Puppet Warp tool is active. Then press the bracket keys to set the depth. The ] key sets the pin forward and the [ key sets the pin backward.
You can also use the Pin Depth icons in the Options bar.
Here’s an example of how this works: