A Diophantine approximation problem with unlike powers of primes

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 $O(X^{1-\frac{83}{144}+2\delta +2\epsilon })$.