If you look for accurate ephemerides on the web, you will find instead of earth, a mysterious object called EMB. This is the earth-moon barycenter, which must be regarded as the object orbiting the sun as a single mass.
To speak of 2006 RH120 as being "perturbed by the moon" is like saying that Apollo 8, e.g., was orbiting the earth and "perturbed by the moon". I notice that the cited article is ambivalent on this point, which is amusing in light of the flap over the planetary status of Pluto. The embarrassing fact is that the earth-moon system is a double planet.
I was keeping it simple. The EMB is always below the surface of the earth, in any case, saying that an object orbiting the EMB is an object in orbit around the Earth, perturbed by the moon is just another way of looking at it. For that matter, planets orbit the barycenter of the solar system, which can be more than two solar diameters from the center of the sun.
For earth satellites, GPS for instance, the second harmonic of the earth’s gravity field, J2, is about 127 times more significant that the moon. The J2 perturbation is on the order of 0.00024 times the J0 component, the effect a point mass equal to the mass of the earht, located at the center of the earth. The effect of the moon is only about 0.0000019 times the J2 component, on average.