The exceptional set for Diophantine inequality with mixed powers of primes

Assume that ${\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4,{\lambda }_5,{\lambda }_6,{\lambda }_7$ are non-zero real numbers , ${\lambda }_1\mathrm{/}{\lambda }_2$ is an irrational number. Let $\mathcal{V}$ be a well-spaced sequence, and $\delta >0$. For any given positive integer $k\ge 5$ and any $\epsilon >0$, we give the upper bound of the number of $\upsilon \in \mathcal{V}$ with $\upsilon \le X$ for which the inequality $$\mid {\lambda }_1{p}_1^2+{\lambda }_2{p}_2^3+{\lambda }_3{p}_3^3+{\lambda }_4{p}_4^3+{\lambda }_5{p}_5^3+{\lambda }_6{p}_6^4+{\lambda }_7{p}_7^k-\upsilon \mid <{\upsilon }^{-\delta }$$ has no solution in primes $p_1,p_2,p_3,p_4,p_5,p_6,p_7$.