On a ternary Diophantine inequality with one prime of the form \(p=x^2+y^2+1\)

Let \(1<c<\frac{127}{113}\) $1<c<\frac{127}{113}$ be a fixed number and N be a sufficiently large real number. In this paper, it is proved that the Diophantine inequality \(\begin{aligned} \left| p_{1}^{c}+p_{2}^{c}+p_{3}^{c}-N\right| <\varepsilon \end{aligned}\) $$\begin{array}{c}\hfill \left|p_1^c,+,p_2^c,+,p_3^c,-,N\right|<\epsilon \end{array}$$ is solvable in prime variables \(p_1,p_2,p_3\) $p_1,p_2,p_3$ such that \(p_1=x^2+y^2+1\) $p_1=x^2+y^2+1$ . This result constitutes a refinement upon that of Dimitrov (Ramanujan J 59:571–607, 2022).