During the past decade, Han and her husband, surnamed Huang, have taken their children to hospitals in Guangzhou and Dongguan for treatment.
Han quit her job to look after the twins two years ago after she found it harder to hire nannies.
The family did not apply to have the boys declared disabled, which would have allowed them obtain government subsidies.
Han's husband said the family was nearly bankrupt after spending all of its savings on the twins' treatment and they saw no sign of recovery.
"I didn't know she would kill them in such a way but I forgive her. I am really tired and I don't want any more tragedies to happen," Huang said.
The family's plight has drawn the sympathy of their neighbors and former colleagues, who jointly wrote a letter to the court pleading for mercy for the mother.
"We went to see her at the detention house. She kept crying. She is so sorry," said a former colleague of Han, who declined to be named.
The court said it will handle the case carefully.
"We received the letter. We also feel sympathetic for the family," a judge from the court told Guangzhou Daily. The judge did not say when the court will hear the case.
"But the case involves murder. We are trying to look for a balance between laws and sympathy in the case," the judge said.
Zheng Ming, a Guangdong-based lawyer, said Han's case should be treated differently to other murder cases.
"Her attempt to kill herself after discovering her sons' death well explains that she was not trying to escape the pressure of looking after her children. It is not simply a murder case," Zheng told China Daily.
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