Assuming you mean awards using AsiaMiles booked entirely on CX, then you can certainly have at least one stopover enroute - I recently ticketed MNL-(x)HKG-(o)SIN-(o)HKG-MNL for 35,000 miles.

If it's a OneWorld Award then from the site: "You can make a maximum of five stopovers, two transfers and two open-jaws at either origin, en-route or turnaround point, subject to airline partners' terms and conditions."