The purpose of the tilling was really to pull up some roots and get to what's underneath. The top is all hard and cracked. We got a ton of roots up and pulled out a lot of weeds. We'll be adding a bunch of new dirt and levelling. I've no doubt that will be a long process, but that's ok. This particular stretch does have full sun. It does have a small crepe myrtle, but that doesn't shade it much. This side faces the south, and being on the corner gets morning, afternoon and evening sun, only a tiny bit of it gets filtered by the crepe myrtle. My only worry is that it'll end up with a greenhouse effect and grow even better under the plastic..... That side has some bermuda already (and we left some areas. Plans for that stretch (which is actually bigger than I said - 37x17, I finally measured it today) are not settled yet. It'll have some bermuda in the center, but we're going to semi-xeriscape. I say 'semi' because we do want some grass, but we'd like to minimize water requirements. I'm going to put a border in around the perimeter, though I haven't decided which plants to put in. The border will be ~2 ft in (a little more around the wrap-around porch to incorporate some bushes already there. Along the fence line I'm still debating. We have an ugly chain link fence which we hate, but aren't planning to remove it any time soon. There's bermuda outside the fence in the front (no sidewalk there), but I was thinking of putting in some ivy-type ground covering (sorry, still learning all the names) which will creep over the ugly fence. On the inside we'll have the mulch with some drought-tolerant plants. There's an ivy-type plant growing elsewhere along the fence, but it's pretty messy and strangles the cedar elm, so I'm not sure I want to keep that. The whole property is 1/4 acre, so we have a fair amount of yard to deal with. As we're still in the midst of interior restorations, the outside is a slow process. It's mainly getting attention now because it's one of the few things I can do with my little one (compared to painting, sanding, hanging sheetrock....)

For comparison, this is Glechoma hederacea growing at my house. 100% guarantee the ID of these pictures - I am an herbalist and do harvest this for herbal purposes. I tried to take a picture of a "patch" that was of similar character with close-ups for the OP to compare to. I do not see the "round objects" in the original photo but Glechoma hederacea do get galls in its stems that are very round. FataMorgana

Campanula rapunculoides, Creeping Bellflower.

Fungi are a fact of life. There are over 100,000 microbial species in your soil of which a very large fraction are fungi. As gardengal said, you will never be free of them and you don't want to be. Whether they are creating mushrooms is something else. You can minimize the mushrooms by proper watering. Watering: Deep and infrequent is the mantra for watering. This is for all turf grass all over the place. Deep means 1 inch all at one time. Put some cat food or tuna cans around the yard, and time how long it takes your sprinkler(s) to fill all the cans. Memorize that time. That will be the time you water from now on. My hose, sprinkler and water pressure takes 8 full hours to fill the cans. Your time will likely be less. I like gentle watering. As for watering frequency, that depends on the daytime air temperature. With temps in the 90s, deep water once per week. With temps in the 80s, deep water once every 2 weeks. With temps in the 70s, deep water once every 3 weeks. With temps below 70, deep water once a month. Note that you have to keep up with quickly changing temps in the spring and fall. This deep and infrequent schedule works in Phoenix and in Vermont, so it should work for you. The reason for deep and infrequent is to grow deeper, more drought resistant roots and to allow the soil to dry completely at the surface for several days before watering again. If it rains, reset your calendar to account for the rainfall.