Does this add security against chosen input attacks, and if so how much?
No, if $R_c$ is public, it is easy to find collisions. The idea is to find a simultaneous collision on both $R$ and $C$.
Here is one approach:
Start with a provisional input $v_i$; set the lower three inputs $v_3, v_2, v_1$ to 0, and everything above that arbitrarily (we won't change those values)
Compute the value $R = H( \sum_{i=0}^{|V|} v_i * R_c^i)$
Now, for any arbitrary constant $c$, we can reset $v_2 = c$, $v_1 = -c(R + R_c)$, $v_0 = cRR_c$
For any such value of $c$, evaluating the function gives us the same value of $R$ (because $cR_c^2 - c(R + R_c)R_c + cRR_c = 0$, consistent with our initial test evaluation), and thus gives the same $H$ value (because $cR^2 - c(R + R_c)R + cRR_c = 0$ and all higher order terms are the same)
By choosing two different values of $c$, this gives us a collision. This same approach would allow computing preimages, should the attacker find that useful.