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One reasonable thing to note is that defining $$f'(x, y)=f(x+1, y+1)+f(x-1y)+f(x-1, y-1)-f(x+1y)-f(x, y-1)-f(x-1y-1)-f(x, y+1)$$ we have that $f'$ is also harmonic in this sense. So we have that arbitrary "derivatives" of harmonic functions are harmonic, though this is much more trivial than in the smooth case.
One reasonable thing to note is that defining $$f'(x, y)=f(x+1, y+1)+f(x-1, y-1)-f(x+1, y-1)-f(x-1, y+1)$$ we have that $f'$ is also harmonic in this sense. So we have that arbitrary "derivatives" of harmonic functions are harmonic, though this is much more trivial than in the smooth case.