  Let $F$ be an algebraically closed field of characteristic $p>0$. In this
paper we develop methods to represent arbitrary elements of $F[t]$ as sums of
perfect $k$-th powers for any $k\in\mathbb{N}$ relatively prime to $p$. Using
these methods we establish bounds on the necessary number of $k$-th powers in
terms of the sum of the digits of $k$ in its base-$p$ expansion. As one
particular application we prove that for any fixed prime $p>2$ and any
$\epsilon>0$ the number of $(p^r-1)$-th powers required is
$\mathcal{O}\left(r^{(2+\epsilon)\ln(p)}\right)$ as a function of $r$.
