  Let $1<c<832/825$. For large real numbers $N>0$ and a small constant
$\vartheta>0$, the inequality \begin{equation*}
|p_1^c+p_2^c+p_3^c+p_4^c-N|<\vartheta \end{equation*} has a solution in prime
numbers $p_1,\,p_2,\,p_3,\,p_4$ such that, for each $i\in\{1,2,3,4\}$, $p_i+2$
has at most $32$ prime factors.
