Is it ok for the gimbal cover to be left on the Mavic Air when it's being powered on? Sometimes I just want to power on my MA so I can mess around in the settings or update firmware.

No gimbal cover also stabilises gimbal for transport you could damage it by turning drone on

I think you always need to remove the gimbal cap before turning on. I've noticed that during the startup process, the gimbal moves to align itself. If the cap it on, then it comes in the way of the startup procedure.

The auto calibration process will be hindered by the clamp.
In the long term that can even damage the gimbal motors.
So no...

There should be a more clearer warning regarding this. I noticed that also  while upgrading FW, batteries, playing around with the settings until a notice popped up on the display like "gimbal motor overheating" or similar since i forgot to take the cover off. So yeah, better be safe than sorry.

It is highly recommended to remove the gimbal cover when the drone is powered on.
The cover is only used to protect the gimbal while in transport

This step should be in the tutorials and quick start... I always do firmware updates for new equipment and being unfamiliar with this model i didn't see the sticker underneath that says remove cover before plugging in device to computer. They could do this now with a pop up on the youtube walkthrough  video without even updating the video. Hopefully my gimbal is OK.
At a minimum they should add it to the pictorial quickstart guide that comes in the box.