Gypsum is claimed to "break up and loosen" clay soils. In areas of the country with high calcium (calcareous) soils, this is a local myth. However, when gypsum (calcium sulfate) is added to clay soils having a high sodium content, the sodium in the soil is replaced with calcium from the gypsum, creating a much more desirable soil condition. The sodium is leached out of the soil with water (irrigation).
High sodium soils most frequently occur in the southwestern United States. Elsewhere, a recommendation of adding gypsum to soil is generally not appropriate. When clay soils are already high in calcium, adding calcium in the form of gypsum is a waste of time and money. The best way to improve disturbed soils in gardens is through copious additions of organic matter. In the case of a lawn, support the turf with water and fertilizer and leave the clippings on the lawn. Over time the grass will add organic matter to the soil and improve its condition and internal drainage.