Stellar clusters around supermassive black holes are systems in which relativity likely plays a role. Currently, only bright stars can be seen in our own galactic center because there is a ton of neutral gas between us and the galactic center that obscures it. As a result, we only have a few "test particles" out of the many stars that actually orbit the black hole at close distances.
Nevertheless, measuring relativistic precession may be possible for a star with one of the closest-known pericenter distances to Sagittarius A* (the central black hole in our galaxy), S2, potentially within the next few years once enough data has been collected.
As to how relativistic effects can affect dynamics of the cluster, the precession induced by general relativity can suppress resonant interactions, including three-body resonances such as the Kozai. Depending on if these sorts of resonances are important compared to other relaxation processes, the relaxation time can increase significantly, resulting in the cluster evolving more-slowly over time. This can affect things such as the rate of mass segregation, tidal disruptions, and production of hypervelocity stars/S-stars.