I heard that at one time the Sahara desert was cropped in wheat for the Roman Empire, and that cropping without adding back to the soil was what caused the desert to form. I also heard that this is going on in the U.S. (in the Great Plains, etc.). Really? Seems odd that the amount of rainfall would differ based on the health of the soil.
For an example of how bad agricultural practices can cause soil loss and end up with dust and sand, see the conditions that preceded the dust bowl in the USA. Certain crops are bad for soil and if not rotated will eventually deplete the soil of nutrients leading to no crops. With nothing holding the soil in place (roots) it will erode. The Sahara has other ill effects going on, such as synoptic scale atmospheric descent (interaction from the Walker Circulation and the Hadley cell) which promotes dry, warm conditions.
I'm not sure that rainfall will vary based on soil health, but one avenue to look into is soil moisture and other sources of water. Rain begins as water vapor and that water vapor comes from evaporation (from bodies of water, entrainment), evapotranspiration (from soil and plants) and advection (blown in with the wind). If the predominant flow does not bring in moisture and you lose the ability to hold water in the soil, this could conceivably locally reduce water vapor mixing ratios and impact precipitation. That is just a guess and it would be worth searching the literature (you can freely download AMS articles older than 2 years old and search them here).