Prime values of f(a,b2) and f(a,p2), f quadratic

We prove an asymptotic formula for primes of the shape f(a,b(2)) with a, b integers and of the shape f(a,p(2)) with p prime. Here f is a binary quadratic form with integer coefficients, irreducible over Q and has no local obstructions. This refines the seminal work of Friedlander and Iwaniec on primes of the form x(2)+y(4) and of Heath-Brown and Li on primes of the form a(2)+p(4), as well as earlier work of the author with Lam and Schindler on primes of the form f(a,p) with f a positive definite form.