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 \(2<c< \frac{68}{33}\) and for any arbitrary large number \(E>0\) , the Diophantine inequality

$$\begin{aligned} \left| p_{1}^{c}+p_{2}^{c}+p_{3}^{c}+p_{4}^{c}+p_{5}^{c}-N\right| <\left( \log N\right) ^{-E} \end{aligned}$$

is solvable in prime variables \(p_1,p_2,p_3,p_4,p_5\) such that, each of the numbers \(p_{i}+2\,\, (1\le i\le 5)\) has at most \(\big [\frac{214467}{136000-66000c}\big ]\) prime factors, counted with multiplicity.