Primes values of $a^2 + p^4$

We prove an asymptotic formula for the number of primes of the shape $a^2 +p^4$, thereby refining the well known work of Friedlander and Iwaniec. Along the way, we prove a result on equidistribution of primes up to $x$, in which the moduli may be almost as large as $x^2$.
Comment: A previous error in the treatment of the bilinear sum has been corrected