If you are talking about those CNY goodies like Bak Kua (on consignment; made by a homebased Malaysian family business) then you have to go to ChinaTown Mart at Upper Riccarton popularly known as Church Corner (opposite the Countdown Supermart). They do have a small bakery section selling Charsiew buns etc on a smaller scale than BreadTalk etc
A couple of competitor Chinese Grocers also operate there including 'Sun Son' and another one (next to Tea'sme bubble tea shop) inside Churchcorner Mall. Sun Son also has a branch at Burnside suburb
At 590, Colombo Street is a recently opened Chinese Bakery by Joyul Restaurant.
There is also Kosco (Korean Supermart) at 277 Blenheim Road with a competitor "Sun Asian Grocery" a stone's throw away. Kosco has several branches in Christchurch, one of which is directly opposite the Central Public Library at Gloucester Street.