What is the effect of moving the centre of gravity ahead or behind the aerodynamic centre for HALE UAVs?
Glider pilots in a competition will try to place the center of gravity at the most rearward position possible, sometimes even further back than permitted. This will make the glider statically unstable, but the performance will not be much improved - this is more psychological than real. Don't discount psychology in competitions: Pilots who feel that they fly an inferior plane will take bigger risks to get ahead and have a much higher chance of spoiling their chances by doing something stupid. Therefore, everyone tries to tweak their plane, even if the tweaking only helps to give a good feeling.

Now to the physics: If the center of gravity (CG) is right above or below the aerodynamic center (AC), the aircraft's longitudinal static stability is indifferent. Move it forward and stability increases, move it further back and the plane becomes unstable. The aerodynamic center is where the angle-of-atttack-dependent lift forces act, and in an indifferent configuration the lift coefficient on wing and tail will be equal. Since the aspect ratio of the wing is normally larger than that of the tail, an indifferent configuration will already produce slightly more induced drag for the same lift than a slightly stable configuration. Details depend on the lift loss at the center of the wing due to the fuselage and the combined vorticity of wing and tail, but as a first approximation the above is valid. Moving the CG further aft only makes matters worse. 

Now it can be argued that the rear CG location helps in maneuvering the aircraft. This might even be helpful with gliders, since they need to maneuver quickly into the center of a thermal, but HALE (high altitude long endurance) UAVs fly mostly straight, so maneuvers will be few, slow and deliberate.

In essence, moving the CG back in a very stable configuration helps, but the optimum is reached already before the CG reaches the AC. Moving it back even further is detrimental to performance and runs the risk of stalling the tail first due to its high loading.

I do not know of any HALE which flies in a statically unstable configuration, and the glider pilots who do will not openly talk about it.

The only application of static instability (CG behind AC) which boosts performance can be found in supersonic aircraft. And the performance improvement is most pronounced in supersonic flight because it reduces the losses from stability of a highly stable configuration. When incorporated from the start, subsonic instability enables the aircraft to use smaller wing and tail surfaces because both will work less against each other at supersonic speed.