A reasonably readable version:
https://en.wikipedia.org/wiki/Transverse_mode
Full-frontal tech:
https://www.edmundoptics.com/knowledge-center/application-notes/lasers/laser-resonator-modes/
The illustrations in those references show that TEM00 mode is a simple Gaussian hump in all directions around the beam axis. The laser can operate in other TEM modes that distribute energy in different patterns, none of which are useful for laser cutting.
The mode is an inherent property of the tube / gases / mirrors, so external tweaks do not affect it. The machine’s alignment and focus can be perfect and the laser will still not cut properly, due to the incorrect distribution of energy within the beam.