Specifically, if I have a presentation $\left<G|R\right>$, and I look at the presentation $\left<G|R,R_1\right>$ is always true that $$\left<G|R,R_1\right>\cong\left<G|R\right>/H$$ for some subgroup $H$ of $\left<G|R\right>$. My intuition tells me yes, but at this time I don't have idea how to prove it. I can do it for specific cases, like if
$$C:=\left<a_1,a_2,\ldots,a_n|\right>,$$ then $$\left<a_1,\ldots,a_n|a_ia_ja_i^{-1}a_j^{-1},\forall i,j\right>\cong C/[C,C].$$
Not directly related to this question, but I was curious about this, because I noticed that if we consider $\left< e|e\right>$ as $$D:=\left<a_1,\ldots,a_n|a_1,\ldots,a_n\right>,$$ then the Cayley graph of $D$ is $\bigvee_{i=1}^n S^1.$ Moreover, $D$ is a quotient of $C/[C,C]$ which has an infinite Cayley Graph, the $n-$cube lattice of $\mathbb{R}^n$, which is a covering space of $\bigvee_{i=1}^n S^1.$ Moreover, both $D$ and $C/[C,C]$ are quotient groups of $C,$ and the Cayley graph of $C$ is the universal cover of the Cayley Graphs of both $C/[C,C]$ and $D$. In addition, $C$ is not a quotient of any group on the generators $a_1,...,a_n.$ So it appears there may be a correlation between quotient groups where the quotients use the same set of generators, Cayley Graphs, and covering spaces.
Yes, there is a group homomorphism $$\phi:\left<G\mid R\right>\to\left<G\mid R\cup R'\right>$$ taking each generator in $G$ on the LHS to the same generator on the RHS. Since each relation in $R$ is also satisfied in $\left<G\mid R\cup R'\right>$ this is well-defined. Then $\phi$ is surjective and so has kernel $H$. Indeed $H$ is the normal closure of the subgroup of $\left<G\mid R\right>$ generated by the words in $R'$.