I'm not sure that
hugelkultur beds are the best for erosion control of this type. If you were going to use them you would want to make sure the water had a place to go so that the hugelkultur beds were not acting like a swale. Otherwise the hugelkultur beds could be blown out causing worse erosion - they are not meant to hold water like a swale. I also question their ability to function as a toe wall without being built a fair bit differently then they are normally.
What seems to be happening to me is that when the hillside was cut to put in the road it messed up the natural slope and now the hill is essentially regrading its self but of
course this is causing problems for you. One option that might help here would be terraces of some sort. You would need to use rocks or something similar that would be strong
enough for this sort of work. I don't have much
experience with terraces but potentially this could help in the worse areas. You would want to support and build up the base of the slope where it meets the road (a toe wall) - I'm guessing this was what you wanted hugelkultur beds to do. I'm just nervous about the ability of a hugelkultur bed to be able to handle this. I think using rocks or something similar would work better but there are also other methods using woven vegetation (see Contour Wattling) that can work but may be difficult in your arid climate - you would have to determine what sort of plants to use.
I also think an issue is the lack of vegetation on the slope. I'm not sure what type of plants would grow best in your arid climate but I think the lack of vegetation is a major cause for the erosion. But without something stabilizing the base of the slope it will be difficult to get vegetation established near the erosion area. It may also be necessary to have additional stabilizing features higher up to give the plants time to get established - this is how the contour wattling method works.
Here are some links with information that might help:
http://www.unce.unr.edu/publications/files/nr/2010/fs1047.pdf
http://www.laspilitas.com/garden/howto/slope.html
http://www.ecy.wa.gov/programs/sea/pubs/93-30/contwattle.html
http://extensionpublications.unl.edu/assets/pdf/g1307.pdf
Hope that helps and gives you some ideas!