I'd suggest you investigate whether you can buy mushroom
manure in your area. Mushroom manure is manure that was used to grow mushrooms commercially. It's spent manure still with plenty of nourishment for your lawn
. It's advantage is that it was steamed 2 or 3 times. It's been used to grow a couple batches of mushrooms. So; many of the weed seeds have germinated and been picked out. The steaming will also kill many weed seeds. It will green up a lawn and any roots
of your trees
and plantings that grow under the lawn. This is a permanent organic improvement to your lawn and the soil.
Spreading this is not as easy as buying a bag of fertilizer as you can't buy a spreader at the big box store. I've bought a pickup load and spread it over a lawn quite a few times. You only need a small covering. A half inch thick. An inch thick covering won't hurt your lawn. You do need to break up the 3 inch diameter clumps that you will find before you spread it. I'd suggest you use a flat shovel not a spade. You simply throw a shovel full over a broad area and on successive throws try to spread it evenly. Small areas can be covered by rocking the shovel so the it falls off the shovel on either side. If this product is available in your area then I'd either use my pickup, a rental, or get it delivered, for about $35 a load in this area. The product is about $35 a yard and you can comfortably get 2 yards in an 8 foot pickup bed. If you use a truck you have the advantage of getting it closer to where your spreading it. Myself tho, I wouldn't drive the truck over the lawn.
Another thing you can do is to spread grass seed
before you spread the manure. A light covering of seed is called overseeding. A heavy layer of seed would be reseeding. If you sow the seed by hand you can do the overseeding/reseeding in one pass by simply sowing as much seed as each area needs. Myself I spread more seed than many do; as I find the amount you save by going cheap isn't worth the effort you've spent in labor.
I've added amendments to the manure. If you have clay I'd suggest you shovel 3 shovels of mushroom manure and one of sand into a wheel barrow. By the time you shovel it out of the wheel barrow and onto the lawn the product will spread extremely evenly. The sand is amazing at changing the tilth. When you buy this it may be quite wet. If it's been raining a lot it'll be much worse than if you buy it after a few days of dry weather. You can also mix some lime into the mixture. There are some on this site that will tell you to never add sand to clay, but they do this on golf courses where it's called topdressing. Also the majority of top soil sold commercially in my area is a shredded mix of clay and sand without any improvements.