First, I want to point out that in general there is no surjection from a direct product of copies of $R$ to an arbitrary module $M$. For example, if $R=mathbb{Z}$ and $M=mathbb{Z}^{(omega)}$ (a countable direct sum of copies of $mathbb{Z}$), then there is no surjection $mathbb{Z}^Ito M$ according to the paper "Extension of a theorem on direct products of slender modules" by John D. O'Neill. [There are probably much simpler examples, but this will do.]
Thus, in general, there is no map $p$, and thus no pushout $N$.
Second, maps defined from $R^{I}$ are notoriously difficult to understand, and often depend on cardinality considerations for the set $I$. In particular, when $R=mathbb{Z}$, things get really weird when $|I|$ is measurable.
That all said, assume $p$ does exist. Since, as you pointed out, $N$ is an injective module, we just need to examine when $N$ is an essential extension of $M$ (viewing $M$ as a submodule). One obvious situation when this holds is when $R$ itself is a (right) self-injective ring, for in that case $N=M$ is already injective. Of course, this case is somewhat trivial since self-injective hereditary rings are already semisimple.
More generally, write
$$N=Moplus E(R)^I/langle (p(x),-i(x)) : xin R^I rangle.$$
For each $ein E(R)^Isetminus{0}$, let $X_e:={rin R : erin R^I}$. The set $E_e={er : rin X_eR}$ is a submodule of $R^I$. Given any element $overline{(m,e)}in Nsetminus M$, we need $overline{(m,e)}Rcap Mneq (0)$. Thus, we need some element $rin X_e$ such that $mrneq -p(er)$. In other words, we do not want $p|_{E_e}$ to extend to a map $eRto M$. Thus, whatever condition you impose will need to restrict what types of maps $p$ are available.
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