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 $\mathrm{\mid }S\mathrm{\mid }\ge l$ has a nonempty zero-sum subsequence $T$ satisfying ${\mathsf{v}}_g(T)={\mathsf{v}}_g(S)$ for some $g\mid S$. We obtain a common upper bound for $c_0(C_q\oplus C_{nq})$ for groups of odd order, where $q$ is prime, and prove $\mathsf{N}(G)\ge r(n_r-1)+1$ when $G=C_{n_1}\oplus \cdots \oplus C_{n_r}$ with $1<n_1\mid \cdots \mid n_r$. If $n\ge 1$ and $p\ge 3$ is the smallest prime divisor of $np$, we obtain $\mathsf{N}(C_p\oplus C_{n{p}^2})=2n{p}^2-1$ and give upper bounds for $\mathsf{N}(C_p\oplus C_{np})$ when $p\nmid n$. We also determine $\mathsf{N}(C_n)$ for two infinite families of cyclic groups arising in earlier work.