Asymptotic Fermat's Last Theorem for a family of equations of signature $(2, 2n, n)$

In this paper, we study the integer solutions of a family of Fermat-type equations of signature $(2, 2n, n)$, Cx² + qᵏy²ⁿ = zⁿ. We provide an algorithmically testable set of conditions which, if satisfied, imply the existence of a constant BC, q such that if n > BC,q, there are no solutions $(x, y, z, n)$ of the equation. Our methods use the modular method for Diophantine equations, along with level lowering and Galois theory.