How would I go about filling the gap shown in the attached photo?
That really needs to have the stepping stones removed and the bed rebuilt. The stones are now not pitching away from the structure, but toward the structure. It will only get worse. Since the steps are pre-formed, this task will be quite easy, and should only take about one saturday to complete. This is something that should be completed before bad weather, so that you can watch and verify that the work you do is holding up. You really cannot fill a gap that wide, because the walkway bed is ever depleting. So the gap will only get wider, thus allowing more water to flow under the steps, and washing away more of the under pining that is holding up the steps and the walkway.
Handy Andy In Mt Airy NC
The steps are hollow and bolted to foundation.
The sagging problem exits along entire walkway as it winds down leftward a good pitch down to my driveway. I'll have to tackle this in chunks, but first need to remedy the underlying cause of sagging before I redo the base.
for now, looking for ideal temporary gap fill to keep the critters out.
One thing that I just thought of that may get you thru until springtime, is the use of clay to fill the gap. It is not permanent, but may do what you need for a little while.
I would look at where and how the water is flowing past the steps and try to divert it for a while.