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Cultured marble is not like Corian or the other so-called solid surface products. The surface coloring is a coating, does not go through the countertop. It can be permanently defaced by deep scratches, excessive scouring or cleaning with abrasive cleansers.
If you have not completely removed the glazed surface by sanding, you may be able to restore some of the sheen by applying a wax product known as Gel Gloss. Though I have not used it specifically on cultured marble, I have had great success with this product on both Formica and plastic shower enclosures that have lost their new surface glow. At the least it will prevent excessive staining of the now-vulnerable surface by typical bathroom medicines and chemicals.
Otherwise, you could try to coat the surface with a few coats of gloss polyurethane varnish, or even paint it with an epoxy base paint. All your prep work will not have gone to waste!
Ah... another of the "costs" of smoking! Cultured marble is a composite product of actual stone dust mixed with a binding product. The surface is sanded incredibly smooth and coated with a special "gel" finish that is durable and waterproof... but not indestructible. The surface is susceptible to scratching, burns and is affected by some chemicals... most notoriously nail polish remover!
You cannot remove the burns with any simple cleaning product. They are permanently etched into the surface of the cultured marble. Actual restoration to original condition is impossible.
However, there are a few repair options. The same folks who do "glazing" on tubs also do vanity tops. They use a special type of paint to give your sink a new surface. You may be able to purchase this paint from a local paint store, but it is quite expensive... perhaps more costly than a new vanity top... and a gallon will be quite a bit more than you need! The only website I have found that sells this product direct to the consumer is:
Other sturdy paints, such as floor paints, will stand up for a while but will not be as durable or long-lasting as the "real thing".
If you want to attempt a spot repair yourself you may be able to mask the burns with a little artistry and a kit from Bath Wizard designed to repair small defects in cultured marble. Their website is http://www.bathwizard.com .