I know the phrase "household chores" is much more common. Tell me if the phrase "housework chores" is also used anywhere in the English speaking world, most probably in Australia or New Zealand or somewhere else?
Household and housework are very different concepts.
Household means a residence , associated property and those that live there Oxford Dictionaries.com.
Housework would be the work to maintain a house in order, particularly cleaning. Oxford Dictionaries.com
Housework is in itself household work, adding chores to housework would be redundant. That would not be good English, and is not accepted usage.
One may have household chores, or housework, but not housework chores.
Apart from household chores, there is another alternative:
domestic chores, the adjective domestic is not redundant and as an expression it is perfectly idiomatic. The term domestic refers to the running of the household (home).
It is the perfect synonym for the compound noun, housework, which already contains the meaning of chore, in work. A mundane, repetitive, but necessary job is called a chore, and household work is precisely that.
2 [only before noun] relating to family relationships and life at home
- Unfortunately his domestic life wasn’t very happy.
domestic tasks/chores/responsibilities etc
Nowadays there is more sharing of domestic chores.
families that can afford domestic help (=help with cleaning, washing etc)
- an organization that supports women facing domestic violence (=violence in a family, especially from a husband to his wife)
Source Longman Dictionary