### A Diophantine inequality with four squares and one $k$th power of primes

Let $k\ge 5$ be an odd integer and $\eta $ be any given real number. We prove that if ${\lambda}_{1}$, ${\lambda}_{2}$, ${\lambda}_{3}$, ${\lambda}_{4}$, $\mu $ are nonzero real numbers, not all of the same sign, and ${\lambda}_{1}/{\lambda}_{2}$ is irrational, then for any real number $\sigma $ with $0<\sigma <1/\left(8\vartheta \right(k\left)\right)$, the inequality $$|{\lambda}_{1}{p}_{1}^{2}+{\lambda}_{2}{p}_{2}^{2}+{\lambda}_{3}{p}_{3}^{2}+{\lambda}_{4}{p}_{4}^{2}+\mu {p}_{5}^{k}+\eta |<{\left(\underset{1\le j\le 5}{max}{p}_{j}\right)}^{-\sigma}$$ has infinitely many solutions in prime variables ${p}_{1},{p}_{2},\cdots ,{p}_{5}$, where $\vartheta \left(k\right)=3\times {2}^{(k-5)/2}$ for $k=5,7,9$ and $\vartheta \left(k\right)=[({k}^{2}+2k+5)/8]$ for odd integer $k$ with $k\ge 11$. This improves a recent result in W. Ge, T. Wang (2018).