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Normally, too much phosphorous does not cause issues as plants have a hard time taking up too much phosphorous in the first place. The soil can be balanced back out some by adding in nitrogen rich amendments like composted manure.

Earthworms introduced into the soil will help aerate and add more nutrients to your lawn. These articles will help as well: https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/how-to-improve-clay-soil.htm
https://www.gardeningknowhow.com/special/organic/compost-for-organic-gardens.htm

If you can't till, you can use lasagna methods for amending the soil. These articles will help:
https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/lasagna-gardening.htm
https://www.gardeningknowhow.com/special/organic/compost-for-organic-gardens.htm

You really need to do a soil test (either a home test or one from the extension service) to tell exactly what your soil might need and how much. This article can help you with knowing what your soil's needs might be:
https://www.gardeningknowhow.com/edible/vegetables/vgen/vegetable-garden-soil.htm

As for your root vegetables, the soil is probably too heavy. You can amend it to make it lighter: https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/how-to-improve-clay-soil.htm

It's possible that what you have in your soil is mychorrizae fungus, a highly beneficial and desireable organism which breaks down organic nutrients for the plant roots to absorb. Mychorrizae is the key to successful bonsai, for instance, and forms a symbiotic relationship critical for potted pines to live in a closed environment. It is a distinctly filamentous (ie. "stringy"), substance which doesn't show above the surface but permeates the root area of the soil. If what you have is a surface organism, it is likely something else altogether. In any case, be it mildew or mold, it will not thrive unless the soil is damp and somewhat shaded.

Since drainage is definitely an issue here that is causing algae to build up in the soil, you may want to consider putting in a raised garden bed instead. Alternatively, you could construct a berm to redirect the drainage to another area. These articles should help: https://www.gardeningknowhow.com/edible/vegetables/vgen/raised-vegetable-gardens.htm
https://www.gardeningknowhow.com/special/spaces/creating-and-using-berms-in-the-landscape.htm
https://www.gardeningknowhow.com/special/spaces/building-a-berm-how-do-i-make-a-berm.htm