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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 binaryquadratic form with integer coefficients, irreducible over $\mathbb{Q}$ and hasno local obstructions. This refines the seminal work of Friedlander and Iwaniecon primes of the form $x^2 + y^4$ and 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 onprimes of the form $f(a,p)$ with $f$ a positive definite form.

Prime values of $f(a,b^2)$ and $f(a,p^2)$, $f$ quadratic