  Let $N$ be a sufficiently large real number. In this paper, it is proved
that, for $1<c<\frac{1193}{889}$, the following Diophantine inequality
\begin{equation*}
  \big|p_1^c+p_2^c+p_3^c+p_4^c-N\big|<\log^{-1}N \end{equation*} is solvable in
prime variables $p_1,p_2,p_3,p_4$, which improves the result of Mu.
