Maximizing difference of a submodular and a modular function
To simplify life, let $\mathcal V = [n] := \{1,2,\ldots,n\}$. For $A \subseteq [n]$, define $h(A):=\sum_{i \in A}c_i$. Note that $h$ defines a modular (i.e additive) set function. Now, suppose there exists $\gamma \in [0, 1]$ such that $f$ is weakly $\gamma$-submodular, i.e such that

$$
\sum_{i \in B\setminus A}f(A \cup \{i\}) - f(A)  \ge \gamma (f(A \cup B) - f(A)),\;\forall A,B \subseteq [n].
$$

Then, greedy maximization produces a subset $A^G$  with $k$ elements such that

$$
f(A^G)-h(A^G) \ge (1-e^{-\gamma})f(A^*)-h(A^*),
$$

where $A^*$ is the $k$-element subset of $[n]$ which maximizes $f(A)-h(A)$.
This is a direct consequence of Theorem 3 of this ICML paper.