  In this paper we show that, for any fixed $1<c<\frac{5363}{3900}$, every
sufficiently large positive number $N$ and a small constant $\varepsilon>0$,
the diophantine inequality \begin{equation*}
|p_1^c+p_2^c+p_3^c+p_4^c+p_5^c-N|<\varepsilon \end{equation*} has a solution in
prime numbers $p_1,\,p_2,\,p_3,\,p_4,\,p_5$, such that $p_1=x^2 + y^2 +1$.
