Housing Australia's latest update to a parliamentary committee hearing showed about 200 more homes had been delivered by the end of August since its previous update in July.
But of the 1607 dwellings delivered, less than 1000 were from projects that had been built from scratch with government money.
Up to 700 were from projects which were already in development but which Housing Australia had then paid money to a housing provider to purchase and turn into social and affordable housing.
Opposition housing spokesman Andrew Bragg questioned how that was helping solve the housing crisis if the homes were already going to be built.
"At the end of the day ... this is not providing additional stock necessarily, because these properties could have been purchased by others," he said on Thursday.
Greens Senator Barbara Pocock was on a unity ticket with Senator Bragg in upbraiding Housing Australia's leadership for their progress.
"As Labor's principal response to the worst housing crisis in living memory, Housing Australia has shown itself to be woefully inadequate to the challenge," Senator Pocock said.
But Housing Australia acting chief executive Sarah Amos said she was confident the target would be met by deadline, with a big rush of completions expected in the final year.
"Over 18,000 homes have been contracted to date, and that is a very strong pipeline towards the target of 40,000 homes by 30 June, 2029," she said.
Even if the target is met, there is no guarantee that the "affordable" homes delivered will actually be affordable to low-income households.
Less than one in five affordable housing rentals goes to a low-income household that can actually afford it, according to a report published by the Committee for Economic Development of Australia on Thursday.
Despite the name, many affordable homes are out of reach for low-income households because rents are often set at a proportion of market rate, rather than a household's income level.
Moreover, broad eligibility criteria mean medium-income households often have access to affordable housing, crowding out those that need it most, the think tank said.
Analysis by CEDA and consultancy group Urbis found only 19 per cent of rentals affordable to a household eligible for social housing actually go to one.
In Perth that figure was one in 10.
Senator Pocock said the government should instead fund public housing directly, like it did in the post-WWII house building boom.
Social housing advocacy group Everybody's Home told the inquiry the government should set a national target of lifting social housing from less than four per cent of total housing stock to 10 per cent.
"We've got a shortfall at the moment of 640,000 public and community homes for the people who are in really severe rental stress, and that is forecast to go up to 940,000 over the next two decades," Everybody's Home spokesperson Maiy Azize said.
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