In your photo there are LOTS of little seedlings, which means there must have been lots of seeds in your lawn or in the compost you put down. You didn't mention the lawn having lots of weeds previously so I wonder about the compost. Where did it come from? Your options are limited while the new grass is just getting started since many broadleaf herbicides can damage the grass seedlings too. If you wait until the grass is established at least 8 weeks, then you may be able to use herbicides. Check the label of any product before using it on new lawns. I agree with kimmsr that correcting soil problems is the way to manage moss in a lawn. Moss is not a competitive plant, as you can see by the very limited root system it has. It grows in lawns where the grass is already doing poorly. This can be cause by lack of soil fertility (the most common cause in my experience), too much shade, heavy soils that drain poorly, and excess acidity (pH below about 5.2). Lack of organic matter isn't normally limiting for grass, but adding organic matter can improve drainage if that is the problem. Have you had a soil test done? You mention that the soil was determined to be acidic. What was the pH?

How big of a garden spot is this? It looks like it would be easier and much quicker to pull up the "good plants" and smother the whole spot for a year. Then uncover a small test spot & check to make sure it's well & truly dead before putting it back into use. Smothering won't work if not done thoroughly, heavy corrugated cardboard with seams overlapping at least 6", no holes in it, covered with at least 4" of heavy organic matter holding it down and completely blocking the light, and for sure covering all of the plant/roots. I think I would use 2 layers of it for this plant, one I couldn't kill when I tried getting rid of the top 6" of soil in a garden about 20 yrs ago. I've never had smothering not kill anything but this particular plant gives me the heebie jeebies. Good luck with whatever you attempt!

Agree. The very best time of the year to seed a northern lawn is in the early fall. Every day after that is worse than the previous day. Once you get past late winter, the results are MUCH less satisfactory headed into spring, late spring, and summer. The later it is seeded the faster it dies out and the faster the crabgrass takes over. At this point all you can do is sit it out. You can practice good lawn care management for the next couple of months and plan to renovate it in late August. Here's what to do until then. 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. Mulch mow at the mower's highest setting. Fertilize now with an organic fertilizer like soybean meal, alfalfa pellets, corn meal, corn gluten meal at a rate of 15 pounds per 1,000 square feet. Don't use any chemical fertilizers, herbicides, or insecticides until later. By allowing the weeds to grow you will be developing the soil and putting roots down into it. These roots are important for your soil to develop the right structure. Then in August do this: Change your watering practices to every day, 3x per day, for 5-15 minutes each time. Do that to sprout all the weed seeds in the soil. After a week of daily watering, spray everything with Round Up to kill all the living stuff. Continue with the daily watering for another week and spray again with RoundUp. This process should clear everything out. Rake up all the dead stuff and you can seed right onto the top of that soil. Roll the seed down or walk on every square inch of it. Then continue watering daily 3x as before to germinate your new grass seed. This approach will give you a pure stand of grass almost guaranteed. When you get 80% germination on your grass seed, start to back off on the watering frequency. Mow when the grass is tall enough to mow at 5 inches or so. At that point look to see if the grass is uniformly dense across the yard. You still will have time to add more seed in the thin areas.

Mmmm, something in the Asteraceae. Maybe a Lactuca. These spread by airborn fluffy seeds so just keeping it cut before it flowers would help. Otherwise just dig them out.