Summary: Given odd, coprime integers \(a, b \; (a > 0)\), we consider the Diophantine equation \(a x^2 + b^{2 l} = 4 y^n, x, y \in \mathbb{Z}, l \in \mathbb{N}, n\) odd prime, \( \gcd(x, y) = 1\). We completely solve the above Diophantine equation for \(a \in \{7, 11, 19, 43, 67, 163 \} \), and \(b\) a power of an odd prime, under the conditions \(2^{n - 1} b^l \not\equiv \pm 1(\text{mod } a)\) and \(\gcd(n, b) = 1\). For other square-free integers \(a > 3\) and \(b\) a power of an odd prime, we prove that the above Diophantine equation has no solutions for all integers \(x, y\) with \(( \gcd(x, y) = 1), l \in \mathbb{N}\) and all odd primes \(n > 3\), satisfying \(2^{n - 1} b^l \not\equiv \pm 1(\text{mod } a), \gcd(n, b) = 1\), and \(\gcd(n, h(- a)) = 1\), where \(h(- a)\) denotes the class number of the imaginary quadratic field \(\mathbb{Q}(\sqrt{ - a})\).