  We prove Bombieri-Vinogradov type theorems for primes with a missing digit in
their $b$-adic expansion for some large positive integer $b$. The proof is
based on the circle method, which relies on the Fourier structure of the
integers with a missing digit and the exponential sums over primes in
arithmetic progressions. Combining our results with the semi-linear sieve, we
obtain an upper bound and a lower bound of the correct order of magnitude for
the number of primes of the form $p=1+m^2+n^2$ with a missing digit in a large
odd base $b$.
