Don't know the solution, but the cause may be that there has been one to many children playing on the area. I base this comment on the fact it looks like you are living in an older neighborhood. (Clue Heaved Sidewalk.)

Good eye, krnuttle. Now that you mention it, the giganticity of the tree is another clue. You might consider replacing that tree with a tree more appropriate to the space it's in, so that it doesn't heave the sidewalk.

Given the amount of shade you have, the only grass that will work for you is a fescue. Fescue plants normally do not fill in bare spots no matter how much fertilizer you use. They have to be reseeded in the fall (NOT the spring) to get dense again. Plan that project for late August after the summer heat breaks. Early seeding gives the grass time to come in and time for you to evaluate whether you need still more seed.

For your hard soil try this. Spray the yard with any clear shampoo at a rate of 3 ounces per 1,000 square feet. An example of a clear shampoo is baby shampoo - you can see through the bottle and the shampoo. Dollar Tree has bottles of shampoo for a dollar. Follow that up with 1/2 to 1 inch of irrigation or rainfall. Give that 3 weeks. If the soil does not soften when moist, then repeat. I sprayed mine in San Antonio back in 2012 and it is still soft when wet. Soils should be firm when dry and soft for a few days when moist - like a sponge. The shampoo seems to work where other, more strenuous, methods do not.

Don't get rid of the Tree. Any thing you put in there will not give you any shade for the as long as you live in the house. (more than 10 years). Consciously, or not it was one of the reasons that you bought house.

Also I suspect the tree is on the street easement, and so is owned by the city.