Cleaning: Charred places are best removed with a scraper. It is important that the scraper is intended for ceramic plates! An ordinary sponge with difficult dirt will not help.
Polishing: There are measures on the market for polishing ceramic tiles, it's hard for me to recall something, but just like above, you just need to look for something that is intended for ceramic plates. It is important that you do not use CIF milk for cleaning, because they have particles that can draw such a disc. And polishing will not have a negative effect on the disc.
Generally, induction hobs often have a ceramic coating, whereas cleaning products that should be used for induction hobs should be adapted to ceramic hobs. There will be no problem.
As for the manufacturer's marks. With cleaners intended for induction hobs, the original markings from the manufacturer should not be erased, although for safety reasons I would advise you to omit these elements. In the case of German induction hobs, the manufacturer's marks are fixed in the board, so in this case, you should not worry about it.
A scraper intended for such plates will not scratch the surface of the plate. You do not have to heat the disc.
Thank you for the answers, I will provide you with the tools and resources you write about and let me know + photos of how I went.
I will tell you only that I am shocked how I would rub the plate had to be some specks of the bar or something I do not know from where and the resulting scratches - I did not think that these plates are so delicate ...
Can you share the record brand? At the moment, in Europe, glass plates are produced from two places France and Germany. One of the better surfaces (if not the best) is of course Schott glass. Some manufacturers fly at cost and order glass from China, then you can compare the used board on European and Asian glass - the difference is huge!
Scratches from "sand" can arise after inaccurately washing the bottom of pots, that if we add some spices like salt, sugar, etc. during cooking, moving the pots, these spices can attach to the bottom.