The burn could be the size of a dime and a circle or it could be the lenght of a cigarett and retangular.
If a circle, I would first try to prune off the burnt areas and see what it looks like. If the carpet has polyesters, then you have a glob of burnt plastic.
If so and if its a circle.
Get a small circular round mellon cutter and cut out the area. The hole should be the size of a penney or so. Now, your down to the pad. Slip into this hole a backer sheet with those bumps of dried glue. Fold it up to get it under the carpet, then with twezers unfold it and have it laid out centered under the hold.
Now, with this same mellon cutter go to the closet and get another good circular patch. the sizes should be the same.
Take this patch and insert it in the old hole and then, be careful, but get a pair of blue jeans or a bath towel and with a hot iron, rub the area with the iron over the towel attempting to melt the glue under the carpet to stick to the new patch. Rub the iron, then check the rug. YOU DO NOT want to cause the rug to turn brown or to burn. If this is not working, then just drop some glue on the bottom of the patch and its there.