The one exception was MESSENGER, which approached Earth from 31 degrees North and left at about 32 degrees South. This path was in a plane containg Earth's centre, which may be a clue. Flybys that are asymnetrical about the equator seem to show the anomaly. Such an effect would not be noticed by the many Earth-orbiting satellites, for their orbits are coplanar with Earth's centre. This strongly suggests that the anomaly is connected with Earth's rotation.
Earth's spin naturally causes a slight acceleration which had been accounted for. This is related to the relativistic frame-dragging effect. If a spacecraft like NEAR is given an extra anomalous velocity, it could mean that part of Earth is rotating much faster. The detailed calculations will take some time, but this anomaly may be another reason to suspect a Black Hole in Earth's core.
Update: The Black Hole is tiny, barely 1/10 of a millimeter in radius but weighing nearly as much as the Moon. It is far too tiny to suck us up, and consumes about as much mass in a year as a human eats. That small amount, partially converted to radiation, keeps Earth's core hot. The Black Hole dates from a time near the Big Bang. When our solar system was just a gas cloud Earth formed around the singularity like a pearl around a grain of sand. Our planet and life would not have formed without this little hole.