  Let $$\gamma^*=\frac{8}{9}+\frac{2}{3}\:\frac{\log(10/9)}{\log 10}\:(\approx
0.919\ldots)\:.$$ Let $\gamma^*<\gamma_0\leq 1$, $c_0=1/\gamma_0$ be fixed. Let
also $a_0\in\{0,1,\ldots, 9\}$.\\ We prove on assumption of the Generalized
Riemann Hypothesis that each sufficiently large odd integer $N_0$ can be
represented in the form $$N_0=p_1+p_2+p_3\:,$$ where the $p_i$ are of the form
$p_i=[n_i^{c_0}]$, $n_i\in\mathbb{N}$, for $i=1,2$ and the decimal expansion of
$p_3$ does not contain the digit $a_0$.\\ The proof merges methods of J.
Maynard from his paper on the infinitude of primes with restricted digits,
results of A. Balog and J. Friedlander on Piatetski-Shapiro primes and the
Hardy-Littlewood circle method in two variables. This is the first result on
the ternary Goldbach problem with primes of mixed type which involves primes
with missing digits.
