The simple explanation that textbooks and the internet say is that "gravitional potential is a scalar quantity hence can be added algebraically".
However, I'm not sure if it is that simple. Take for example, at point O where it is on the horizontal axis connecting the centre of the Moon and the centre of the Earth such that the gravitational field strength at that point is zero. Any point above or below point O will have a net gravitational field strength that is the vector sum of both Earth's (gearth) and Moon's gravitational field strength (gmoon). This vector sum will be smaller than (gearth)+(gmoon) but larger than (gearth) or (gmoon) alone.
Now back to the definition of gravitational potential at a point in a gravitational field, which is the work done (by an external agent) per unit mass in moving a mass from infinity to that point. This work done will be the integral of the gravitational field strength with respect to the distance away from the source of the field.
Therefore, since the gravitational field strength at point O due to the Moon and the Earth is zero, and the fact that gravitational field strength above and below point O isn't (gearth)+(gmoon), hence the work done per unit mass in moving a mass from infinity to point O will not be as simple as adding the gravitational potential due to the Moon and Earth algebriaically.
Am I correct?