The point $P_i=f'(R_i),$ where $f'(x)=v_0 x+ v_0 x^2+v_0 x^3$ is a polynomial,.Normally one would also include a constant term in the polynomial. I will assume that for simplicity.
In any case, since the set of polynomials over $\mathbb{F}_p$ is closed under addition and scalar multiplication, what you are doing would directly correspond to using a linear code, in this case a Reed-Solomon code if the $x$ term was not there. So I will define $f(x)=f'(x)/x,$ and work with that.
It is very easy to mount a second-preimage attack on this method. First compute any point in the nullspace of the mapping $x\mapsto f(x),$ call it $N_f\subset \mathbb{F}_p.$ Thus find any value $R'$ such that $f(R')=0.$ This can be easily done by trial and error. Since the polynomial $f(x)/x$ has degree 2, it has lots of roots in $\mathbb{F}_p.$ Having done this, any point of the form $R_0+\alpha R'$ with $\alpha\neq 0,$ will also satisfy $P'(V)=f(R')=x,$ if $P_0(V)=f(R_0)=x.$
Technical Note: Since you have included an extra multiplicative term of $x$ in your definition, and the difference of the values need not have an $x$ term, you are looking at an affine subspace as opposed to a linear subspace which is the reason for your difficulty in attempting this problem.
