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Unfortunately, fungi such as this one, are very difficult to identify without a microscope. What we can say for certain is that the fungus prefers woody, lignin rich substrate. Depending on what species, it can sometimes try and eat your plants.

The best way to get rid of the issue is to treat with fungicides. Removing soil will not ensure that it has not crept outside of the area. It will infect the new soil, as well.

https://www.gardeningknowhow.com/garden-how-to/info/using-fungicides-in-garden.htm

You can just cover this area with clear plastic and let it bake. You don't need to remove the soil.

https://www.gardeningknowhow.com/garden-how-to/projects/how-to-solarize-garden-beds-to-eliminate-garden-pests-in-the-soil.htm

That is a type of lichen. It is going to be difficult to pin down the species when dried, but it might be a species of Cladonia or Peltigera. Some in the species Peltigera are toxic. This is only if you eat them, though.

Removing them isn't straightforward. This will mean removing the moss and any low growing plants that hold moisture to the ground. Elevating the area to keep it more dry is a good idea. Treating with a fungicide can help, as well.

You have to rake those up or mow them, then pick up the pieces and discard in trash. There must be fungi below ground on some decaying wood of some kind. These are their fruiting bodies. The soda will not help. You will have it till the dead wood or roots below ground are completely decomposed.

https://www.gardeningknowhow.com/ornamental/fungus-lichen/environmental-benefits-of-fungi.htm

https://www.gardeningknowhow.com/ornamental/fungus-lichen/mushroom-identification.htm

I suspect the compost is not fully composted and the fungi is helping to decompose the added organic material. Moisture and humidity will also cause it to grow. Cut back on the frequency of adding the compost and see if that helps. If you need to get it out of your kitchen composter you can start a compost pile in an out of the way spot in your yard.