I fear that the staircase has reached its end and that it's time to demolish it and cast a new one.
But it's certainly possible to patch and repair. I would just mix the concrete drier (earth-damp), like a mortar, and toss it into the cavity. A bucket of gray stone can also be handy to fill out with as well. Just press it into the concrete as you go, it also provides reinforcement. Make sure to "prime" the surfaces with runny concrete before you start so that the new concrete mortar adheres to the old casting. Don't build too high so that the concrete sags. You might want to build this in two or three stages, starting from the bottom and building upwards.
Stop with this firm concrete 2 cm inside the final surface. When the concrete has cured for a day, you can arrange a layer for a nice finish.
I guess it's water seeping into the construction that causes frost wedging. There should be drainage possibilities.