As the warm iron didn't work, the next relatively non-invasive method you could try uses industrial alcohol (aka Meths), a fine white cotton cloth (e.g., old white handkerchief) and a wad of cotton wool.
The method is to make up a squash ball size± wad of cotton wool, reasonably loosely packed, but not too loose, and wrap this in the cloth: place the wad of cotton wool in the centre of the cloth and pull the cloth's edges up and twist these together so that you have a crease free roughly dome shaped underside. Effectively you've made a polisher's rubber. Next unwrap the cotton covering cloth, and pour some alcohol over the cotton wool, not soaking, but enough that a gentle squeeze pushes out some alcohol. Wrap up the cotton cloth as before to recreate the rubber.
Now the tricky bit. Squeeze the rubber to get a bit of alcohol on to the covering cloth, just a smidge. Hold the rubber in your hand and start a swinging pendulum action above the white spot. Keep lowering your swinging hand until the cloth just lightly brushes the surface of the polish, and work your way over the damaged surface a small spot at a time, keeping the surface of the cotton cloth just barely damp with the alcohol.
Be patient, cautious, and gentle. Monitor progress, take breaks, have another go if it looks like the technique is improving things, but more work seems to be required. This may work, it may not, and that depends on the finish you're working with, e.g., it works on shellac, nitrocellulose finishes and usually pretty well on pre-cat lacquer - I can't recall for sure what ercol used to use on their furniture before they went all water borne, but I'm pretty sure it wasn't shellac. You'll know if it works, or not, just by looking at the results after probably just a few minutes, perhaps a bit longer.
If that fails, it may be time to get relatively aggressive and drastic with your repairs, but it sure as hell helps if you already know what you're doing, ha, ha. Slainte.