$\mathbb Z_k$-harmonic function that distinguishes two vertices of a graph

Let $$G$$ be a simple, undirected, connected graph on $$n$$ vertices, and let $$A$$ be an abelian group. A function $$f:V(G)\rightarrow A$$, on the vertices of the graph $$V(G)$$, is said to be $$A$$-harmonic if for any $$v\in V(G)$$, $$\deg(v) f(v) = \sum_{w\in N(v)} f(w)~,$$ where $$N(v)$$ is the set of neighbours of $$v$$ and $$\deg(v) = |N(v)|$$.

Question 1: Let $$A = \mathbb Z_k$$ be the ring of integers modulo $$k$$. Given two distinct vertices $$v,w\in V(G)$$, I want to know when there is a $$\mathbb Z_k$$-harmonic function $$f$$ such that $$f(v)\ne f(w)$$.

When $$A=U(1)$$, I know the complete answer using the injectivity and universal property of the Abel-Jacobi map on the graph. (See Chapter 3.5 of [1] for definition and properties of this map.) The answer is as follows: there is a $$U(1)$$-harmonic function $$f$$ with $$f(v)\ne f(w)$$ if and only if there are at least two paths from $$v$$ to $$w$$ in $$G$$.

I am not sure what to do when $$A=\mathbb Z_k$$ except for some very special graphs. For example, say $$G$$ is the cycle graph on $$n$$ vertices. Then such an $$f$$ exists if and only if the number of edges in either path from $$v$$ to $$w$$ is not divisible by $$\gcd(n,k)$$.

I have looked for any literature on this question and found that a $$\mathbb Z_k$$-harmonic function is also called a balanced vertex-weighting [2], or a conservative vertex colouring [3]. However, these notions don't seem to be commonly studied, and they don't discuss when two vertices can be separated by a $$\mathbb Z_k$$-harmonic function.

Question 2: Is there an equivalent way of phrasing the above question that's more commonly studied, say, a $$k$$-colouring-type problem?

[1]: Corry and Parkinson, Divisors and Sandpiles (2010), http://people.reed.edu/~davidp/divisors_and_sandpiles/mbk_draft.pdf

[2]: Berman, Bicycles and Spanning Trees, SIAM Journal on Algebraic Discrete Methods 1986 7:1, 1-12

[3]: Lamey, Silver, and Williams, Vertex-Colored Graphs, Bicycle Spaces and Mahler Measure (2014), https://arxiv.org/abs/1408.6570