The quaternary Piatetski-Shapiro inequality with one prime of the form $\mathbf{p}={\mathbf{x}}^{\mathbf{2}}+{\mathbf{y}}^{\mathbf{2}}+\mathbf{1}$

In this paper we show that, for any fixed $1<c<967\mathrm{/}805$, every sufficiently large positive number $N$ and a small constant $\epsilon >0$, the diophantine inequality \begin{equation*} |p_1^c+p_2^c+p_3^c+p_4^c-N|<\varepsilon \end{equation*} has a solution in prime numbers $p_1,p_2,p_3,p_4$, such that $p_1=x^2+y^2+1$.