How can I remove these spots?
I would try the plumber gel with baking soda as it might work. I'm going to try it this weekend as I have the same problem. I might put some Vaseline on the metal drain as I'm not sure if it could damage it.
Make a slurry of hydrogen peroxide and baking soda. Apply to the stains and let it dry. Wipe and rinse.
You can try making a paste with baking soda, and peroxide and let it sit then lightly rub in and wipe off. I fear it's the chemical reaction that happens to plastic resin. countertops when something is left on it too long and it doesn't get air or light. Moving the items off and let them be exposed to light, it may fade away or even out.
I don't know what "avonite" is but here is how to remove yellow stains from a counter: