I do similar repairs. Make sure to dig out every bit of the organic material and as much of the mud as you can. Make a "V" out of the wash so it will pack better. Go back with a well-mixed batch of dirt and clay. Pack it in with small lifts as mentioned with your bucket full of dirt for weight. Widen up your backfill so you can pack a wider area.
If your soil is sandy bring in a load of clay to mix in with the back fill.
If your soil is sandy bring in a load of clay to mix in with the back fill.