Over an arbitrary field $\mathbb{F}$, Harbourne conjectured that $$I^{(N (r-1)+1)} \subseteq I^r$$ for all $r>0$ and all homogeneous ideals $I$ in $S = \mathbb{F} [\mathbb{P}^N] = \mathbb{F} [x_0, \ldots, x_N]$. The conjecture has been disproven for select values of $N \ge 2$: first by Dumnicki, Szemberg, and Tutaj-Gasińska in characteristic zero, and then by Harbourne and Seceleanu in odd positive characteristic. However, the ideal containments above do hold when, for instance, $I$ is a monomial ideal in $S$. As a sequel to (arXiv:1510.02993), we present criteria for containments of type $I^{(N (r-1)+1)} \subseteq I^r$ for all $r>0$ and certain classes of ideals $I$ in a prodigious class of normal rings. Of particular interest is a result for monomial primes in tensor products of affine semigroup rings. Indeed, we explain how to give effective multipliers $N$ in several cases including: the $D$-th Veronese subring of any polynomial ring $\mathbb{F} [x_1, \ldots, x_n]$ $(n \ge 1)$; and the extension ring $\mathbb{F} [x_1, \ldots, x_n, z]/(z^D - x_1 \cdots x_n)$ of $\mathbb{F}[x_1, \ldots, x_n]$.

For Version 2: 19 pages, material in several sections of the paper have been re-written and re-grouped. The preliminaries for divisor class groups and for toric algebra have been updated to include results from Robert M. Fossum's book and, e.g., Fulton's book, respectively. We updated the bibliography to include some additional references. To appear in the Journal of Algebra