I looked closely at your photographs and concur with Missing Rosie's thoughts. If it is truly on a very SMALL sloping hill and not something that constitutes a hazard I think I would take the time to learn about the property, learn about basic erosion issues (library, google, local government sources and extension services) and just focus on the basics for now. I would fill in the missing soils and plant something like daylily's or something that forms a good deep root system that would help anchor the soil. It is hard to recommend anything more specific, not knowing your climate, soils, or sun/shade needs. You could also consider a French Drain
https://en.wikipedia.org/wiki/... type of system around the patio if you think it is warranted. Basically you only know there is a problem right now. You are not familiar with the property enough to know if this was a one time event or a new problem or an old problem that is likely to re occur. Living tissue (plants) coverage is natures way of controlling water erosion. In such a small area in which it is NOT a hazard to property below or threatening your home, it is worth taking the time to study exactly what the problem is, when it occurs and this leads to the solution. I would not spend the money now on a new deck or patio if this one is in good shape and the erosion can be controlled. It is much more fun to spend money on your new home making it suit you!
