In Lightroom, a simple fix is to use the Adjustment Brush on the shiny areas with subtle settings: slightly lower exposure/brightness and reduce clarity. A large feathered brush helps the correction blend naturally, and keeping the adjustment mild avoids making skin look flat or fake.

For prevention, the real issue is usually hard, direct flash creating specular highlights. If possible, reduce or diffuse the flash. With compact cameras or pop-up flashes, even a simple diffuser in front of the flash can soften the light and noticeably reduce shininess. People have used thin white paper or DIY diffusers for this, but be careful with heat and make sure anything placed over the flash is safe.

Another traditional solution is using makeup/foundation to cut facial shine before shooting.

So: Lightroom can improve it after the fact, but softer flash and less reflective skin are the best fixes at capture time.