Let me summarize. We take a basis $x$, $y$ of $E$ and the characteristic is $p$. There are two cases where there is a surjective map $E\otimes S^r(E)\to S^{r-1}E$.

The first case is when $r=p-1$. Then the surjective map does not split. Its kernel is spanned by $x\otimes x^r$, $y\otimes y^r$, $(x\otimes x^{r-j}y^j-y\otimes x^{r-j+1}y^{j-1})$, with $1\leq j\leq r$.

The second case is when $r+1$ is not divisible by $p$. Then the map splits and its kernel is spanned by $x\otimes x^r$, $y\otimes y^r$, $(r+1)x\otimes x^{r-j}y^j-j(x\otimes x^{r-j}y^j-y\otimes x^{r-j+1}y^{j-1})$, with $1\leq j\leq r$.

All this can be checked by tedious computations in in Jantzen's framework of $G_sT$-modules. No further theory is required.

Let me summarize. We take a basis $x$, $y$ of $E$ and the characteristic is $p$. There are two cases where there is a surjective map $E\otimes S^r(E)\to S^{r-1}E$.

The first case is when $r=p-1$. Then the surjective map does not split. Its kernel is spanned by $x\otimes x^r$, $y\otimes y^r$, $(x\otimes x^{r-j}y^j-y\otimes x^{r-j+1}y^{j-1})$, with $1\leq j\leq r$.

The second case is when $r+1$ is not divisible by $p$. Then the map splits and its kernel is spanned by $x\otimes x^r$, $y\otimes y^r$, $(r+1)x\otimes x^{r-j}y^j-j(x\otimes x^{r-j}y^j-y\otimes x^{r-j+1}y^{j-1})$, with $1\leq j\leq r$.

All this can be checked by tedious computations in Jantzen's framework of $G_sT$-modules. No further theory is required.