都正确，物主代词或定冠词可加可不加。例如：

do (one's) homework《现代英语综合大辞典》

do (the) housework《朗文当代英语词典》

以下例句均摘自于美国当代英语语料库（COCA）:

Danny goes to his room to do homework. 

I am alone on the bus and I use the time to do my homework and read.  

Are you raising your son to learn how to do housework? 

Then, when I come back home, I have to do my housework, everything.