Let $F$ be a finite field, $n, k, m$ be natural numbers. I give you $m$ vectors $c^{(1)},\ldots,c^{(m)}\in F^n$. I ask for polynomials $p_1,\ldots,p_n$ on $k$ variables over $F$ such that the system of polynomial equations $p_i(t_1,\ldots,t_k)=c^{(j)}_i$ for $i=1,\ldots,n$ is satisfiable for every $1\leq j\leq m$.

Such polynomials can be found with degree $1$ if $k=n$: just take $p_i^{(j)}(t_1,\ldots,t_{k}) = t_i$. Can one find such polynomials when $k=n^{\epsilon}$ for a small $\epsilon>0$ and with degree depending only on $1/\epsilon$?