Voters in Virginia and New Jersey this week sounded a serious warning to Democrats, key players in the Biden administration and Congress said on Sunday: the party needs to get things done or it faces disaster in midterm elections next year.

The energy secretary, Jennifer Granholm, said “we thank God” something was done on Friday night: a $1tn infrastructure deal sent to Joe Biden’s desk by the House.

Three days after Democrats lost a race for governor in one state Biden won comfortably and barely held the other, House centrists and progressives managed to come together, with some Republican support.

Biden hailed a “monumental step forward” and a “blue-collar blueprint to rebuild America”. He also said “the one message that came across” in Virginia and New Jersey was: “Get something done.”

Ron Klain, the White House chief of staff, echoed his boss, telling NBC’s Meet the Press the American people “wanted to see more action in Washington. They wanted to see things move more quickly, and three days later, Congress responded.”

But Democrats punted again on the second half of the president’s domestic agenda, the 10-year, $1.75tn Build Back Better package to boost health and social care and to seek to mitigate the impact of the climate crisis.

Granholm told CNN’s State of the Union: “I think that the Democrats in the House got the message very loud and clear. Pass the bill and pass the second part too, because these contain things that everyday people care about.

“The governor of Michigan, Gretchen Whitmer, ran on the phrase ‘Fix the damn roads’. And that’s what this bill does. It fixes the damn roads. It fixes the bridges. It gets broadband to real people. It fixes your homes so that they’re not leaking energy.”

Granholm also said the infrastructure bill did not help with childcare and other “costs of living for real people”. That, she said, is the job of Build Back Better, which now awaits analysis by the Congressional Budget Office, a measure demanded by centrists.

The New Jersey centrist Josh Gottheimer told CNN he and his allies wanted to make sure the bill was “fiscally responsible and paid for”. He said he was confidant it would pass but dodged when asked repeatedly if his group would vote no if CBO analysis differed from White House and congressional estimates.

In New Jersey, the Democratic governor, Phil Murphy, won by an unexpectedly narrow margin. Taking a page from Donald Trump’s playbook, the Republican Jack Ciattarelli has refused to concede.

In Virginia, Terry McAuliffe, a former governor, suffered a devastating defeat by Glenn Youngkin, a businessman who kept Trump at arm’s length while campaigning on culture war issues including the place of race in education.

Asked if Youngkin could have been beaten had major legislation been passed in Washington before election day, Mark Warner, a Democratic senator from Virginia, told CNN: “I wish the House would have moved earlier.”

Warner also said voters needed to be told what was in the Biden bills, rather than what they cost. The bills’ cost is regularly condemned by Republicans – and by Joe Manchin, the West Virginia Democrat who remains a key obstacle in the Senate.

The White House adviser Cedric Richmond told Fox News Sunday Manchin was “a lot more conservative and everybody sees that but he’s been a willing partner to come to the table with constructive dialogue. And we’re confident in where we will go with our Build Back Better framework. We’re optimistic we’re going to get it done. And the truth is we need to get it done.”

Richmond also rejected Republican claims that increased spending will add to inflation. Granholm said the administration saw current inflation as “transitory”.

Larry Hogan, the governor of Maryland and a Republican moderate with presidential ambitions, told CNN Biden had “nearly snatched defeat from the jaws of victory”.

The infrastructure bill “should have been an overwhelming win back in August”, Hogan said. “And I think [Biden] should not have let it get sidetracked by the progressives in the House. I think that was bad for Joe Biden. I think that was reflected in the election results because I think they misread the mandate.

“You know, Joe Biden won a very narrow election by winning swing voters and they’re not where the progressive caucus is, I can assure you, and the vast majority of Americans are not for the second bill.”

Progressives contend otherwise. In tweets on Saturday, the Washington state congresswoman Pramila Jayapal highlighted news from the Cop26 climate summit in Glasgow and said: “This is EXACTLY why we need the Build Back Better Act. We will deliver climate action – for our communities, future generations, and our planet.”

She also retweeted the Rev William Barber, the leader of the Poor People’s Campaign. He said: “My prayer is that Congress will keep its word and vote to pass Build Back Better, because if not, that political betrayal will be a political crime and integrity breach.”

Such a failure, Barber said, “would abandon over 140 million poor and low-wealth people who make up 43% of the nation and 30% of the voting population”.

That, he said, “could split the Democratic party in ways that may be irreparable”.