The presence of moss indicates a soil problem, mostly that the soil does not have enough organic matter in it. Power raking is not going to solve the "problem" nor will aerating and fertilizing. You need to add organic matter to the soil, lots of organic matter, and then look at how much sunlight reaches that area.
Once the soil is properly amended then grass could grown there and no moss will.

I don't what area you are located in but where I live in the Florida panhandle, I recommend if there is a persistant presenced of moss in particular areas, turn that area into a flower bed if feasible. Grass will never grow where it's suitable for moss. Combination of too much shade, poor drainage, possibly compacted soil.

I live in eastern Washington state and, unlike Seattle, the climate is dry. The lawn had been fine until an evergreen tree next to the main patch got big, creating a lot of shade. I have some cow manure I could spread on the areas. I have already fertilized with organic chicken poo.

Keeping in mind that most all of the cattle, cow, steer manure commercially available comes from Confined Animal Feeding Operations where the cattle are fed grains, not grass, and many disease pathogens grow in the conditions created by those grains, and those cattle are kept in small enclosures so they are knee deep in manure, commercial cow, cattle, steer manure may not be a good choice.
Vegetative waste is the best thing you can add to that soil, tree leaves, garden waste, compost.