Primes from sums of two squares and missing digits

Let ${\mathcal{A}}^{\mathrm{\prime }}$ be the set of integers missing any three fixed digits from their decimal expansion. We produce primes in a thin sequence by proving an asymptotic formula for counting primes of the form $p=m^2+{\mathrm{\ell }}^2$, with $\mathrm{\ell }\in {\mathcal{A}}^{\mathrm{\prime }}$. The proof draws on ideas from the work of Friedlander-Iwaniec on primes of the form $p=x^2+y^4$, as well as ideas from the work of Maynard on primes with restricted digits.