  Let $1<c<\frac{1787}{1502}$ and $N$ be a sufficiently large real number. In
this paper, it is proved that for any arbitrarily large number $E>0$ and for
almost all real $R \in (N,2N]$, the Diophantine inequality
$$|p_{1}^{c}+p_{2}^{c}-R|<(log N)^{-E}$$ is solvable in prime variables
$p_1,p_2$ such that, each of the numbers $p_{1}+2,p_{2}+2$ has at most
$[\frac{79606}{35740-30040c}]$ prime factors, counted with multiplicity.
Moreover, we prove that the Diophantine inequality
$$|p_{1}^{c}+p_{2}^{c}+p_{3}^{c}+p_{4}^{c}-N|<(log N)^{-E}$$ is solvable in
prime variables $p_1,p_2,p_3,p_4$ such that, each of the numbers
$p_{i}+2(i=1,2,3,4)$ has at most $[\frac{93801402}{35740000-30040000c}]$ prime
factors, counted with multiplicity.
