Soil under slab (sidewalk) washed away and is now causing cracks down the middle
Any concrete that is wet enough to flow into that space is probably too wet which weakens its strength. "Pouring" concrete is a misnomer. When it is properly mixed it has to be placed or pushed into position.

I don't think you a footing anyway you just need to replace the soil that was washed away. Usually concrete slabs for roadways are supported by packed gravel/sand mix with some clay content called road gravel. This mix is also used for county gravel roads. 

"Road gravel" from a local supplier could be packed under the slab with a make-shift ram using a 4x4 post or other such device. You could also rent a jack-hammer with a compactor attachment. It sounds like a lot of hard work but rammed earth walls are an accepted way of building your own house with simple tools and a strong back.

Good luck!