What is weight-climb speed?
A cruise weight-climb is a climb segment during the cruise phase of the flight to achieve a higher flight level that was previously not reachable due the weight of the aircraft. 

Long range aircraft cannot climb as high initially because of the weight of the fuel they carry. Their operating ceiling is power limited. As the aircraft burns off weight during the flight, the drag decreases and gradually excess power becomes available for climbing. 

This recommendation is not to start climbing to the next level until a vertical climb speed of 500ft can be maintained at the new level, at the initial cruise speed after that weight-climb. 

For example, if the the cruise starts at FL270, and one wants to climb to FL290 for a cruise at Mach 0.80, the weight-climb should not start before the aircraft can maintain a 500 ft/min climb at Mach 0.80 at FL290.