Can I edit the motion path? Can I change the motion path only by editing the line in the motion path?
I want to change the path by editing the line created by the motion path



Instead of following the white line I want my bone to follow the red one
No, you can not interact with the Motion Path (the white dotted line with yellow keyframe indicators). It is merely a visual reference.

You can edit the motion of the object ...


by setting additional keyframes.
by changing the keyframe interpolation or bezier types in the dopesheet 
by changing the F-Curves of the animated channels in the graph editor
by animating it along a path with a Follow Path Constraint




Editability of the motion path may be achieved through the addon Motion Trail Addon:


Blender Addon List
BlenderArtist post



Motion Path Addon (gif)