Eggroll, Welcome to the forum! I love tthe screen name!

My first though was-- "Oh no! another DIY'r wanting to repair a serious problem on a dam"!!! In your case with dirt work experience, You could possibly do it. You will want to excavate the washed out area completely. Find a source of GOOD clay to backfill in 6" lifts and compact each layer with at least a good Jumping Jack compactor. Tamping it just won't get the job done. The clay also has to have the proper moisture content to compact properly--it can't be too moist or too dry.

After you get the repair re-filled, tied into the existing good sections of the dam properly and compacted, use some topsoil to recover the repaired area, compact it fairly well also and re-seed it right away to prevent eroision.

Soil amendmants or a retaining wall will not help you here. A repair like this takes some heavy equipment and proper reconstruction or the net big rain could cause the entire repair fail at once---taking out awhatever is below it.

I would suggest using a full size (15K#) track hoe at least for the repair.