On a ternary Diophantine problem with mixed powers of primes

Let $1<k<33\mathrm{/}29$. We prove that if ${\lambda }_1$, ${\lambda }_2$ and ${\lambda }_3$ are non-zero real numbers, not all of the same sign and that ${\lambda }_1\mathrm{/}{\lambda }_2$ is irrational and $\varpi$ is any real number, then for any $\text{eps}>0$ the inequality $ \bigl|\lambda_1 p_1 + \lambda_2 p_2^2 + \lambda_3 p_3^k + \varpi \bigr| \le \bigl(\max_j p_j \bigr)^{-(33 - 29 k) / (72 k) + \eps} $hasinfinitelymanysolutionsinprimevariables$p_1$,\mathrm{.}\mathrm{.}\mathrm{.},$p_k$.