What do you use to polish wood plinth?

I would agree with Elizabeth that someone tried to fix a marking and ruined the topcoat in the process. One solution is to remove everything from the plinth so that you are left with just the wood to deal with. Take 0000 steel wool and remove the balance of the to topcoat to an even dull finish. Then remove all dust with tack cloth and re-apply a topcoat finish of your choosing. Try to pick a satin or a semi-gloss to help conceal any imperfections that may exist. Apply as many coats as you wish till you reach the look you are most happy with. Just repeat the steel wool and tack cloth between each coat. I would say that a spray poly would be the easiest to work with or you may want to use a waterbased clear finish. But from your description, I think refinishing is the inevidable solution. Trying to patch it in will create more frustration and may cause further damage, requiring restaining and then refinishing. You can find many woodworking sites that will walk you thru these steps to make it seem less daunting.

If Howard's Feed-N-Wax didn't work I would just adhere a nice Hello Kitty sticker to the spot and call it a day.

Do not use steel wool with water-based finishes. The fine metal dust (even after using a tack cloth) will oxidize.