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 $(2,2n,n)$, $C{x}^2+q^k{y}^{2n}=z^n$. We provide an algorithmically testable set of conditions which, if satisfied, imply the existence of a constant $B_{C,q}$ such that if $n>B_{C,q}$, 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.