The Diophantine equation $x^p+1=py^2$

Cao has recently proved that, subject to a certain condition on the odd prime $p$, the equation $x^p+1=py^2$ has no solutions in positive integers $x$ and $y$, provided also that $p\equiv1\pmod 4$. It is the object of this note to remove this restriction, and to provide a simple self-contained proof.