On the Diophantine equation Cx²+p²ᵐ=yq

We classify the primitive positive solutions of Cx²+p²ᵐ=yq under the hypotheses that $C>3$ is odd and square-free, $p,q$ are odd primes, p C, q h(-C), and y is odd. The solutions form six explicit families: one for arbitrary q and five exceptional families with $q=3$, three of which arise from half-integral elements when C≡38. The proof combines ideal factorization with Lehmer sequences and shows that representations with second coordinate v=pʳ, $r>0$, contribute only when $q=3$.