| Summary: | Let $ N $ denote a sufficiently large real number. In this paper, we prove that for $ 1 < c < \frac{104349}{77419} $, $ c\neq\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.
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