solni, the situation with the original poster was that it would appear randomly and since F1 is assigned to open it I played a hunch but your situation seems different in that it only occurs when you click start.
I guess an easy thing to try is to remove Help and support from the start menu.
Here is my menu with Help and Support...

But if you right-click on the start orb you can pick properties and customize the menu.

Uncheck Help. And click OK and then Apply. Does this resolve it or does it still open up? Does something else start opening instead?
What is the make/model of your mini? Are you using a touchpad rather than a mouse?
The reason I ask, another thing you might try if you have a touchpad ...
Normally if you double tap a touchpad then it acts as a single click. I found it quite easy to accidentally double tap so I disable that option, I much prefer having to use the actual left click button. What I'm thinking is possibly you get a double tap on your menu? If you go to your mouse options you can look for a "tap to click" setting and uncheck that. See if that helps.