Let $q$ be a power of a prime $p \neq 3.$ We
characterize the following
two sets of polynomials: $M(q)=\{P \in {\bf F}_{q}[t]$ such that $P$ is a strict sum of cubes in ${\bf F}_{q}[t]\}$
and $S(q)=\{P \in {\bf F}_{q}[t]$ such that $P$ is a strict sum of cubes and squares in ${\bf F}_{q}[t]\}.$
Let $g(3,{\bf F}_{q}[t])=g \geq 0$ be the minimal integer such that every $P \in M(q)$
is a strict sum
of $g$ cubes. Similarly let $g_1(3,2,{\bf F}_{q}[t])=g$ be the minimal integer such
that every $P \in S(q)$
is a strict sum of $g$ cubes and a square. Our main result is:\begin{itemize}
\item[i)]
$4 \leq g(3,{\bf F}_{q}[t]) \leq 9\,\,\,$ for $q \in \{2,4\}.$
\item[ii)]
$3 \leq g_1(3,2,{\bf F}_{q}[t]) \leq 4\,\,\,$ for $q =4.$
\end{itemize}