Hi, what is the best route options if I am going from Hong Kong to Dunhuang, Urumqi and Turpan?

I'm afraid you have to pass through Lanzhou if heading for DH. Upon arrival at Lanzhou, you can take a train to DH, but the train takes around 14 hours. If you fly there, it only takes about 1h and 40min.

Then to Urumqi, you can also fly there and it takes around 1h and 25min, but I remember the flight is only in service on Tuesdays and Saturdays.

From Urumqi to Turpan, there are over 20 trains in service and it takes around 1.5h.

Thanks. Is it a good time to visit the desert attractions in late Dec? How is the weather then, and do the attractions look the same as other times in the year?

It is not good to travel there in Dec, because winter there is cold and dry. The average temperature in Dec is around -13 to 0C. The attractions are the same, but the scenery is not so beautiful. if possbile, it is suggested to go there in Oct.