Senator Jacqui Lambie has issued an emotional plea to strip politics out of efforts to establish a national housing fund to kickstart building new social and affordable homes.

The crossbench senator was on the verge of tears on Thursday as she implored the Greens to use their balance of power to vote through the government’s $10 billion Housing Australia Future Fund.

Labor’s signature legislation is designed to build 30,000 new social and affordable houses over five years, but has run into significant hurdles in both the lower house and the Senate.

Senator Lambie and her Lambie Network colleague Tammy Tyrrell secured a guarantee of 1200 homes for every state and territory under the Albanese government’s housing plan. But the Greens continue to hold out for more immediate investment and a national rent freeze.

“We are falling behind because for every one we build, we’ve got nearly bloody 50 more on that waiting list,” Senator Lambie told the chamber on Thursday.

“Let’s get the program started so we can get moving, so we don’t have as many [homeless] people out there, especially our children, that next generation.

“I don’t want to see them starting their lives while living in a tent. We cannot hold this up another day.”

Senator Lambie said she knew it would take time to get the materials and labour to build the homes and while it wasn’t perfect, “people out there need a roof over their heads”.

Drawing on her own experience, Senator Lambie said her mother would have been “absolutely paralysed” if the family was forced to live in a tent instead of having the safety net of social housing to instead move into.

“Please, for you people over here, that think you have a social conscience?” she asked the Greens.

“Do you really want to keep playing with people’s lives? Do you really?”

The government and the Greens remain in a stand-off, with repeated moves by Labor to force the fund to a vote in the Senate failing.

The interest from the $10 billion fund, up to $500 million a year, will go towards 30,000 new social and affordable houses over five years.

The government’s leader in the Senate, Penny Wong, accused the minor party of playing politics and pulling media stunts by opposing a fund that would help homeless people and those fleeing domestic violence.

In an acrimonious debate on Thursday, Senator Wong took aim at the Greens housing spokesperson, Max Chandler-Mather.

“Let me talk to you about the Greens spokesperson on housing,” she said.

“You know, he’s had a taste of the media spotlights. He’s had a taste of the media spotlight.

“Your spokesperson on housing is now prioritising media attention from stunts and obstruction over housing for women and kids fleeing domestic violence. How shameful.”

Senator Wong later withdrew the comments about the Griffith MP after Greens senator Nick McKim said she was “well out of order”.

“She is not only wrong, she is very clearly showing that Mr Chandler-Mather is right under the skin of the government,” Senator McKim said.

Even after withdrawing the comments, Senator Wong returned to her criticism.

“You are standing in the way of the biggest investment in housing in a decade and you may yell as much as you like but you are blocking funding for social and affordable housing in this country,” she said.

The Greens hit back on Thursday, saying Labor’s fund didn’t guarantee any money towards social housing if it made a loss in a given year and the legislation did nothing for renters.

“The government is bringing a bucket of water to a house fire,” leader Adam Bandt told ABC radio ahead of Thursday’s Senate vote.

“The rental crisis will get worse over the next couple of years … We’re saying we’re prepared to work with you on your housing package, but it’s not enough to just have a tick-the-box approach.”

After Thursday morning’s heated debate, the Coalition, One Nation and the Greens united to block debate on the housing fund, successfully passing a motion to control the day’s agenda in the Senate.

A vote on the Housing Australia Future Fund was pushed out until mid-June.

Opposition housing spokesman Michael Sukkar said the bill contained “aspirations with no guarantees”. The government should not be borrowing billions of dollars amid surging inflation and a large deficit, he said.

Independent senator David Pocock wants amendments to the bill to include an ability to periodically increase the $500 million annual cap on disbursements from the fund, and indexation at 2.5 per cent each year.

He said it made no sense for the government to have to change legislation every time the investment fund had a good year.

But he has said he would not block the fund from passing.

– with AAP