Definetely hydrophobic soil, given your location most likely a combination of decades of droughts, heat waves and poor watering practices.
The best solution for a vegetable patch is to use surfactants, aka wetting agents. There are many types available and by the looks of it you need a "penetrant", meaning a surfactant that will help the water penetrate into the soil.
Penterra should be easy to find and relatively cheap: it used to come in 2.5, 5 and 55 gallon formats. It needs to be applied every 2-4 weeks during the Summer to see results.
Alternatively the soil need to be broken and worked with a lot of manure or home-made compost. A motorized tiller would help a lot, as long as it's not one of those small toys big box stores sell or the usual Chicom junk. Something like this:
https://orecamerica.com/produc...
Failing that a lot of sweat will do just fine.
