Yes, you can polish your plastic glasses frames.
First of all, you shall clean the glasses frames and remove all grit and dirt on the glasses frames.
Second, place a small amount of non-abrasive toothpaste on an soft cloth, and use the cloth on the scratches of your glasses frames to buff it out. Then wash out any toothpaste on your glasses frames.
Third, check your glasses frames, of there is scratches on your frame, repeat the toothpaste again until you completed all of the necessary buffing and removed all of the scratches.
In my opinion, the first thing is to buy some certain cleaners that are specifically made for polishing plastic glasses frames. You should put the cleaner on the surface of the frames. Then, remember to use the tools or cloth to clean the frames. Try to be careful and avoid scratching the frames.
I don't know your glasses' scratches is on which position of the frames. If it is on the end of the frames or inside, I think it would be unnecessary care about, because others can not see it and would not affect its beauty, could not affect your vision either.
If it have affected the glasses' beauty, you can choose to use a toothbrush brush with toothpaste lightly, seeing whether can clean out or not. If no, I suggest you can go to the shops you bought the glasses and ask if they have the corresponding treatment measures. If they do not have a way to the case, you can only buy the same frames, and also can save more money than you change a pair of new glasses.