The thing to do is to remove the layer of damaged finish with a mild abrasive such as an automotive rubbing compound. Cigarette ash and toothpaste may work some times, but usually are not strong enough to work well.
An automotive paint store has many different grades of rubbing and polishing compounds from which to choose. It is best to try a mild one first, then going on to a stronger one if needed. Unfortunately, trial and error is the only way to find out what you need.
Be careful to check the results often. The finish may not be thick enough for much rubbing. If you do get through to the wood, then the top will need refinishing. It is not possible to tell ahead of time if the damage is through all layers of finish, once again trial and error is the only way to find out.
After rubbing out the white ring, the sheen is different from the rest of the top, so the entire top will have to be done for a uniform look.