Labour has been locked in a bitter power struggle since the 23 June Vote to leave the European Union, with critics of Corbyn saying he did not work hard enough to persuade the party's voters to back remaining in the bloc.
The 67-year-old Corbyn is automatically on the ballot as the current leader and he has strong grassroots support but Smith is backed by 162 of the party's 230 MPs including former leader Ed Miliband.
As Mr Corbyn - who is facing a leadership challenge from Owen Smith - complained that there were "many people in this country struggling with insecure jobs", he was met with raucous laughter from Tory MPs.
Later Mr Corbyn said: "It is not up to me who is a candidate or the MP in any constituency".
The struggle to define the party's identity has seen more than 180,000 people signing up as registered supporters during a 48-hour window this week, paying 25 pounds each ($33) in order to be able to vote in the leadership election.
Former shadow cabinet minister Mr Smith warned the party could be "destroyed" and "consigned to history" unless it could unite.
Branch secretary Terry Brough said: "The branch condemns the coup against the proud trade unionist and socialist, Jeremy Corbyn - the democratically elected Leader of the Labour Party; and we find the evidence to support the accusation that Jeremy is an electoral liability to be flimsy at best!"
Speaking to Sky News, Corbyn said he was "very surprised and actually very disappointed" with the accusation that under his leadership bullying and harassment had been neglected. Another Labour MP said: "If we don't win [the leadership] this year, we will do it again next year and, if necessary, the year after".
He spoke out after Jeremy Corbyn angered MPs, who have called on him to stand down, by suggesting that they could be ousted in 2018. "Labour now has 600,000 members, which I understand makes us the largest left-wing party in Europe".
A party official said that the total number is likely to change as Labour goes through its applicant checking "political vetting" process. I know they will have got involved for the same reasons I did, because they want to tackle inequality and be part of building a fairer Britain and world.
Unveiling his first policy proposal, he said the next Labour government would require every employer with more than 20 staff to publish "equality pay audits" in a bid to eliminate the gender pay gap and other pay discrimination by shaming bosses.
"Because politics has to be about bringing people in, and I think we've done that spectacularly well".
He opened his comments by recounting the Labour's electoral achievements in local elections, by-elections and mayoral elections, and said his campaign would not only set out how Labour can confront social ills in opposition, but also spell out 'some of the measures the next Labour government will take to overcome them'.