Hi Gilded,
Sounds like you need to open it up and take a look. There are two components to the mouse buttons; the electrical button and the mechanical part that depresses the electrical button. It appears something has gotten out of whack with the mechanical button, so use this guide to open it up and take a look.
[guide|159768|How to Open Your Logitech MX Master - iFixit Repair Guide|new_window=true]
The two mouse buttons on the circuit board are being pointed out by the red arrows.
[image|3009770]
Look opposite those spots on the top of the cover to find which part presses down on those white bars to click the mouse. Compare the left and right sides and that should let you figure out what's going on with that left button that it's not working right.
You're always welcome to post pictures of what you find once you've got it open; if you do that then we can see too and perhaps may have suggestions on what needs to be done.
[guide|21499|Adding images to an existing question - iFixit Repair Guide|new_window=true]
Good luck; let us know what you find!