Thanks for the question and sorry to see it is such a bad situation. (This problem is due to overgrazing and too few trees following villagization in the Arusha region. There are some sources on this from the case study of Murray-Rust, D.H. 1973. (see bottom of answer))
So, this is a systemic problem that requires long term changes which require terracing, sand dams and tree planting. A series of videos shows this and goes through the reduction in soil loss that can be achieved with each of these. Overview https://www.youtube.com/watch?v=DmPLG... Sand dams: https://www.youtube.com/watch?v=Yjzcf... and trees https://www.youtube.com/watch?v=yAZRK...
These approaches can transform the landscape in a relatively short time.
Sonia Sachs of the Millennium Village Project just emailed me news of a transformative project in Koraro in Ethiopia that changed the flow of water using Gabion boxes/baskets which is wire mesh around large stones and other interventions such as percolation ponds, irrigation ponds, dug wells, dams and piping. You can get introduction from one of their videos https://www.youtube.com/watch?v=4AS-A... (thanks Sonia!). This was a project initiated in 2005 and certainly making such big changes is a multi-year effort. We are working on getting more data on the success at Koraro to share with you.
Looking at your picture I can see that this water seems to flow from a concentrated stream before branching out (so about 300m before the erosion). Laying down a barrier like tree trunks, rocks, sticks, mud would cause the water to collect and go into the soil and reduce the waterfall erosion effect further down.
But you ask about cover crops so need an immediate solution (besides the dam that would help). I doubt if cover crops will be good enough. The rains are continuing so you wont sow them in the rains and by the time the rains finish you wont need them. (As a side note: Cover crops are an amazing approach to mainatining soil moisture, promoting higher nitrogen and increased oxygen: great video here on their use in USA setting which can also be very dry http://www.youtube.com/watch?v=nWXCLV... )
For Africa, the FAO do have training manual on cover crops in Africa and the relevant details start on page 83. I also attach a picture of a table from that publication (page 95). http://www.fao.org/ag/ca/AfricaTraini...
In planting cover crops you have to decide what to use them for. Some are good for animal feed and others are not (and even bad for animals. See the table).
What are farmers doing there? Raising cattle? Growing crops?
If you can provide more information on what the land is being used for that could help people out there point you to the best cover crop.
good luck and keep sending more information so that we can help.
("Soil Erosion and Reservoir Sedimentation in a Grazing Area West of Arusha, Northern Tanzania." In A. Rapp et al., "Studies of Soil Erosion and Sedimentation in Tanzania." BRALUP Monograph 1. University of Dar es Salaam, Dar es Salaam). If you need this let me know.
Table from Page 95 of FAO Training Manual of Cover crops in Africa