Summary: In this note, we show that if \(p\) is an odd prime and \(p\ne (2^m\pm 1)/d^2\) for any integers \(m\) and \(d\), then the equation \(x^n+2^ky^n=pz^2,\ k\geqslant 2\) has no solutions in nonzero pairwise coprime integers \(x, y, z\) and prime \(n\) with \(n>p^{8p^2}\).