How can I delete the blue mouse travel path without deleting the mouse icon?

As Lilybiri correctly said, the mouse path only appears in Edit mode, not after publishing.

However, the whole purpose of the mouse movement is to show a simulation of a user interacting with the original software application.  If the mouse does not move from one slide to the next, it can be difficult for someone watching the simulation to know what will happen next.  You need to 'lead the eye' from one spot to the next where interaction will take place.  Just using the mouse click alone will not be enough (in my experience).