Let a prime $p\equiv 1(\text{mod}3)$ and $z$ be non-cubic in ${\mathbb{F}}_p$. Gauss proved that the number of solutions of equation

$$x_1^3+x_2^3+z{x}_3^3=0$$

in ${\mathbb{F}}_p$ was $p^2+\frac{1}{2}(p-1)(9d-c)$, where $c$ was uniquely determined and $d$, except for the sign, was defined by

$$4p=c^2+27d^2,c\equiv 1(\text{mod}3).$$

In 1978, Chowla, Cowles, and Cowles determined the sign of $d$ for the case of 2 being non-cubic in ${\mathbb{F}}_p$. In this paper, we extended the result of Chowla, Cowles and Cowles to finite field ${\mathbb{F}}_q$ with $q=p^k$, $p\equiv 1(\text{mod}3)$, and determined the sign of $d$ for the case of 3 being non-cubic.

Let ${\lambda }_1$, ${\lambda }_2$, ${\lambda }_3$, and ${\lambda }_4$ be non-zero real numbers, not all negative. Suppose that ${\lambda }_1/{\lambda }_3$is irrational and algebraic, $\delta >0$, and the set $\mathcal{V}$ is a well-spaced sequence. In this paper, we prove that, for any $\epsilon >0$, the number of $v\in \mathcal{V}$ with $v\le X$ such that the inequality

$$\begin{array}{r}\left|{\lambda }_1{p}_1+{\lambda }_2{p_2}^2+{\lambda }_3{p_3}^3+{\lambda }_4{p_4}^4-v\right|<v^{-\delta }\end{array}$$

has no solution in primes $p_1$, $p_2$, $p_3$, $p_4$ that does not exceed $O(X^{1-\frac{83}{144}+2\delta +2\epsilon })$.