Create mouse/touch path and object follow C#

I am trying to figure out how to draw a path with the mouse and or touch and have a player or game object follow that path. I have a script that reads the mouse position and records it, but the problem I am running into is that sometimes the mouse is too fast for the script. What I mean is that the script does not accurately record the mouse position if the mouse is being dragged too fast. Is there another method? I have been working on this for days now. I am trying to create a game like Yoshi's Fruit Cart. https://www.youtube.com/watch?v=-TB6Wf7HlJ0