Huygens' principle in curved spacetimes

It generally does not work in curved spacetime. There is a quite thick book almost completely devoted to study this issue by P. Günther: Huygens' Principle and Hyperbolic Equations. Some discussions can be found in Friedlander's book about the wave equation in curved spacetime. A necessary condition for the validity of the Huygens principle is that the spacetime be an Einstein space. For Ricci-flat spacetimes there are only two cases, one is Minkowski spacetime the other is a space containing plane gravitational waves.

There are also implications regarding the characteristic Cauchy problem...

It is possible for spacetime curvature to scatter and reflect light. The most obvious case of this is gravitational lensing. It's probably best to just solve the wave equation for the underlying light against the correct metric than to appeal to a simplifying principle like Huygen's principle.