Evaluating $\sum_{x=0}^{p-1}\left(\frac{x^3+a}{p}\right)$

Question

We know that

$$\sum_{x=0}^{p-1}\left(\frac{x^2+a}{p}\right) = -1$$

for any $p\ge 3$, $p\nmid a$.

Is there any way to evaluate the following?

$$\sum_{x=0}^{p-1}\left(\frac{x^{\mathbf{3}} + a}{p}\right)$$

Some empirical computation shows that the value fluctuates a bit, so I'm not sure if there's a pattern.