A quaternary diophantine inequality by prime numbers of a special type

Abstract: 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.