The ternary Goldbach problem with a prime with a missing digit and primes of special types

Abstract: Let $$\gamma^*:=\frac{8}{9}+\frac{2}{3}:\frac{\log(10/9)}{\log 10}:(\approx 0.919\ldots):,\ \gamma^*<\frac{1}{c_0}\leq 1:.$$ Let $\gamma^*<\gamma_0\leq 1$, $c_0=1/\gamma_0$ be fixed. Let also $a_0\in{0,1,\ldots, 9}$. In [23] we proved on assumption of the Generalized Riemann Hypothesis (GRH), that each sufficiently large odd integer $N_0$ can be represented in the form $$N_0=p_1+p_2+p_3:,$$ where for $i=2, 3$ the primes $p_i$ are Piatetski-Shapiro primes - primes of the form $p_i=[n_i^{c_0}]$, $n_i\in\mathbb{N}$ - whereas the decimal expansion of $p_1$ does not contain the digit $a_0$. In this paper we replace one of the Piatetski-Shapiro primes $p_2$ and $p_3$ by primes of the type $$p=x^2+y^2+1:.$$