PCB Design - Copper Pour
Capacitance. 

You will need to wait (to charge the capacitance) before making leakage measurements. As long as the "leakage" stops, this is just basic physics and in no way a defect on the board. 

Removing the copper pour will reduce the capacitance, but in many applications that capacitance is a good thing, rather than a problem! Indeed, about a quarter of the components fitted to the board later, are only there to increase the capacitance between power and ground planes!

Discuss this "problem" with the customer : he will probably be glad to have it!