On a binary Diophantine inequality involving prime numbers

Let $N$ denote a sufficiently large real number. In this paper, we prove that for $1<c<\frac{104349}{77419}$, $c\ne \frac{4}{3}$, for almost all real numbers $T\in (N,2N]$ (in the sense of Lebesgue measure), the Diophantine inequality $|p_1^c+p_2^c-T|<T^{-\frac{9}{10c}\left(\frac{104349}{77419}-c\right)}$ is solvable in primes $p_1,p_2$. In addition, it is proved that the Diophantine inequality $|p_1^c+p_2^c+p_3^c+p_4^c-N|<N^{-\frac{9}{10c}\left(\frac{104349}{77419}-c\right)}$ is solvable in primes $p_1,p_2,p_3,p_4$. This result constitutes a refinement upon that of Li and Cai.