How can I repair the foundation and plaster coating in my old home's foundation?
The material underneath the plaster looks remarkably like the clay lump that my farmhouse in Suffolk, England was made from.  Clay lump is sandy-gravelly-clay+straw+lime, formed into blocks which are dried, and then built into walls with clay+lime mortar.  

My guess is that your bump is not actually clay lump, but it is very likely to be an earth-based construction material (cob for example).

With luck, the plaster is lime plaster rather than a gypsum plaster (lime plaster is much more flexible).  Mixing lime plaster from lime putty (hydrated calcium hydroxide) is quite easy (particularly if you have a cement mixer).  Use a ratio of 2 parts sharp sand to one part lime putty, reinforce with finely teased hair, and you should be good to go.

You may well find that the aggregate in the plaster is your local subsoil (quite possibly from the basement you are standing in!)  If you have some subsoil exposed, it would be entirely reasonable to use it.