Yes.
For a sufficiently small point mass the geodesic is due to the background field. If the "falling" mass is large, its geodesic is due to all fields, including its own contribution to the overall net field, via its effect on the configuration of the other masses.
For a sufficiently small point mass the geodesic is due to the background field. If the "falling" mass is large, its geodesic is due to all fields, including its own contribution to the overall net field, via its effect on the configuration of the other masses.