A better approach would be to lightly rough up the soil where you want to fill in with new seed. Put out fertilizer, seed, press, tamp or roll the seed into the soil. Then, if you want, you can put a LIGHT covering of compost on top to help it stay moist while it is germinating. I'd say maybe a quarter inch of a light compost at most.

Keep it moist for three weeks to a month, then start tapering off the watering.

Note, it will take more watering in the summer to keep grass planted in the spring going when it gets hot.