I would buy grass seed intended for your area from a local dealer, not a big box store. You want a mixture for your area not a mixture that will grow anywhere in the world.

Cover the bare spots with that good seed and then cover it with mushroom compost that you buy in bulk from a local dealer. A 1/4 to 1/2 inch thick layer will keep the seed from drying out till it sprouts and be a semi permanent soil improver. An inch thick layer won't be too much, just more work.

Knowing where your from and how big an area may be helpful in solving your problem.

Rake the topsoil in.

Agree with raking to distribute the topsoil they dumped, and agree with buying good seed from a local place. Don’t know your location, but late summer / early autumn is a great time to seed here.

I had some work done on my house years ago that involved ripping out a section of lawn. I raked the topsoil fairly smooth, sprinkled on good grass seed, then covered it with burlap that I bought at a local garden center. The burlap was inexpensive and came in a long roll — like 3’ wide and 100’ long — so I cut it into long strips to cover the area. Then I kept it watered and it was growing lush in days. It worked so well that all the neighbors and contractors were asking me how I did it! The great thing is that the burlap keeps the seed in place, keeps the birds from eating the seed, keeps the area moist for seed to sprout quickly, and then it quickly composts and disappears!

That's brilliant. I wonder how much burlap costs these days per square yard?

Kevin, if you EVER get the urge to topdress your yard again, please write in first and give me a chance to dissuade you. There is almost never a reason to add more soil. Just to spoil the secret, start using organic fertilizer at least once a year, and you will never feel like adding topsoil.