Hi
Even though it is fairly open weave, it WILL affect the air distribution pattern, downstream of the mesh.
Initially I did not think this was too critical, keeping in mind how thin the screen is. However having replaced my MAF only last Friday I still have the faulty unit and decided to examine just how it is setup.
First of all, both the plastic "grill" in front of the mesh and the mesh itself are "keyed" this means that they must fit in a specific (rotational) position. This position looks to be at 45deg to the maf wire. So I therefore assume that the air distribution patten does have relevance. My thoughts are though, that the plastic grill is of more importance than the mesh as the plastic grill is much "thicker" than the mesh.
If your mesh is partially missing, then more air will bypass the central sensing tube and the MAF reading will be incorrect.
This is not logical. More air
will not pass the sensor! The engine will STILL only draw in the amount of air it needs. The displacement of air or the distribution pattern may change slightly, but there will not be an increase in air mass.
How did the mesh break? If it is constantly disintegrating for some reason then yes, I would remove it. If it has been broken due to mishandling/accident and therefore not likely to break further then I would leave it alone.
I would remove the mesh and run an adaptive-reset in the ECM
From my limited, though recent venture into the realm of MAF sensors

I would not recommend the adaptive reset. I chose to do this adaptive reset during my MAF foray and it made the already poorly running engine MUCH worse. It partly recovered after a few kilometres of driving, but due to the faulty MAF sensor, the values could not recover properly. Replacing the sensor IMMEDIATELY improved the engine operation and I did not have to do anything with the RCL (Rovacom Lite) to "adjust" the running conditions - it just adapted.
So if you remove the mesh or not, the engine will still run as it did prior to any "mesh fiddling" on your part (provided you don't break anything or do something silly

)