Might depend on how secure your run is....if something can get into the run, then yes.
Just realized the building the coop is against has wide open doors?
Might need apron on that side of coop too.
If you stack rocks or pavers around the coop/run, animal will likely start digging at outer edge of rocks/pavers and get underneath them.
With mesh, the animal will start digging at bottom of wall of coop/run and hit the mesh,
it won't back up and get under edge of mesh to go under(provided the...
Ehhh....I'd start with the shavings. Observe and adjust.
If you find the rain comes in on same side consistently, might want to add a solid half or full wall there. Could also maybe extend roof.
Both of you....
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Hard to say what bedding would work best,
might depend on how much rain comes in the sides.
Might think about an apron around all 3 sides to keep anything from digging under the floor and living there.
Good examples of anti-dig apron installation...