No, the relation is not as simple in this case. For example, if $k = 3$ and $p = 7$, the three different cubic Gauss sums are roots of $y^{3} - 21y - 7$, and the three roots of this polynomial do not have equal absolute values.

In general, the $k$ different Gauss sums $g(a,k)$ (where $a$ runs over representatives of the cosets of the $k$th powers in $\mathbb{F}_{p}^{\times}$) will be roots of a polynomial of degree $k$, and the coefficient of $x^{k-1}$ in that polynomial will be zero, because
$$
\sum_{a \in \mathbb{F}_{p}^{\times}/\left(\mathbb{F}_{p}^{\times}\right)^{k}} g(a,k)
= k \sum_{n=0}^{p-1} e^{2 \pi i n / p} = 0.
$$
When $k = 2$, this forces both quadratic Gauss sums to be roots of a polynomial of the form $y^{2} - a$ and this makes the quadratic Gauss sums be negatives of each other, but this is no longer the case when $k > 2$.

Gaussian periods, see en.wikipedia.org/wiki/Gaussian_period – Seva Jul 8 at 20:13