The government expects most of the unlawfully raised debts paid under the robodebt scheme will be refunded by November, with the $721 million to start heading back into the pockets of welfare recipients from July 1.
For those who are currently receiving Centrelink payments, a spokesman for minister Stuart Robert has confirmed that the refunds won't be counted as income, meaning the payments, for some worth thousands of dollars, won't reduce their ongoing payments.
The refunds won't be siphoned off to pay other social security debts people may have.
While hundreds of thousands of people were contacted by the government in recent weeks to notify them they could be eligible members of the class action brought by Gordon Legal, the 373,000 people who will be repaid are a smaller subset of those contacted so far.
The Gordon Legal class action required everyone who had paid a debt under the income compliance program since 2015, while the refunds are only going to those whose debts were raised either fully or partly using income averaging.
As more details emerged, the debate continued over whether the government should apologise for the program, which Prime Minister Scott Morrison refused to do. Instead Mr Morrison said there was "great regret" about the robodebt debacle, and shrugged off questions about people who suicided after receiving debt notices.
There are also calls for a parliamentary or judicial inquiry into the program.
Almost 400,000 people will be repaid $721 million in unlawfully raised debts under the government's income compliance initiative, which started in 2015. The program used an income averaging process to compare a welfare recipient's fortnightly reported income to Centrelink against their annual income as reported to the tax office.
The government "paused" the program when it lost a test case in the Federal Court last year, and agreed to pay back the falsely calculated debts late on Friday afternoon in the face of a class action.
Asked about the government's handling of the program on Monday, Mr Morrison sought to spread responsibility to the Labor party.
"The income averaging principle is one that has been followed by Labor and Coalition governments for a very long period of time," Mr Morrison said.
"Over the course of dealing with this issue, that principle was not something that could be relied upon. That doesn't mean those debts don't exist. It just means that they cannot be raised, solely, on the basis of using income averaging."
Since Friday's announcement, three ministers and now the prime minister have refused to apologise for the program, citing the class action that is still before the courts.
"This has been a very difficult project for many but the Government is putting it right," Mr Morrison said.
"We're still working through some legal procedures right now, so I think the time for those sorts of statements are at another time, not right now."
While the use of income averaging to raise debts against welfare recipients has been used before 2015, the program introduced by the government in 2015 led to the automation of the debt raising, and forced the onus of proof onto current and former welfare recipients to prove they didn't have a debt, instead of the other way around.
Attorney General Christian Porter said on Sunday there would be a statute of limitation" on people who had debts raised using averaging before 2015.