Is there any correlation between barometric readings and the phases of the moon?

Yes, the phenomenon is known as the lunar atmospheric tide. Although the sun has a greater tidal influence on the atmosphere, the moon does also play a role.

Georg Balthazar Neumayer, who worked in Australia in the late 1800s, discovered the lunar tidal effect on the atmosphere. It has been shown that the effect is stronger in the tropics than at higher latitudes. The effect is pretty small. If I remember correctly, the size of the lunar tidal oscillation in the tropics is around 0.1 millibar. A millibar is a measure of pressure, and the average surface pressure on the earth is around 1013 millibars. Strong tropical cyclones might cause local surface pressures to drop as much as 50 to 60 millibars. However, there generally isn't a lot of surface pressure variability in the tropics, especially compared to the mid-latitudes, so the lunar tidal effect is detectable.

-- Michael Bell