On a diophantine inequality involving prime numbers of a special form

Let N be a sufficiently large real number. In this paper, we prove that for \(2<c< \frac{68}{33}\) $2<c<\frac{68}{33}$ and for any arbitrary large number \(E>0\) $E>0$ , the Diophantine inequality \(\begin{aligned} \left| p_{1}^{c}+p_{2}^{c}+p_{3}^{c}+p_{4}^{c}+p_{5}^{c}-N\right| <\left( \log N\right) ^{-E} \end{aligned}\) $$\begin{array}{c}\hfill \left|p_1^c,+,p_2^c,+,p_3^c,+,p_4^c,+,p_5^c,-,N\right|<{\left(log,N\right)}^{-E}\end{array}$$ is solvable in prime variables \(p_1,p_2,p_3,p_4,p_5\) $p_1,p_2,p_3,p_4,p_5$ such that, each of the numbers \(p_{i}+2\,\, (1\le i\le 5)\) $p_i+2(1\le i\le 5)$ has at most \(\big [\frac{214467}{136000-66000c}\big ]\) $[\frac{214467}{136000-66000c}]$ prime factors, counted with multiplicity.