Additive bases and the zero-sum invariant $\mathsf{N}(G)$ for groups $C_p\oplus C_{np}$

Let $G$ be a finite abelian group, and let $c_0(G)$ be the least integer such that every regular sequence over $G$ of at least this length is an additive basis. Let $\mathsf N(G)$ be the least integer $l$ such that every weak-regular sequence $S$ over $G$ with $|S|\ge l$ has a nonempty zero-sum subs