  In this paper, we consider the equation \[ x^2 - q^{2k+1} = y^n, \qquad q
\nmid x, \quad 2 \mid y, \] for integers $x,q,k,y$ and $n$, with $k \geq 0$ and
$n \geq 3$. We extend work of the first and third-named authors by finding all
solutions in the cases $q= 41$ and $q = 97$. We do this by constructing a
Frey-Hellegouarch $\mathbb{Q}$-curve defined over the real quadratic field
$K=\mathbb{Q}(\sqrt{q})$, and using the modular method with multi-Frey
techniques.
