By Elizabeth Piper and William James

LIVERPOOL, England (Reuters) - Jeremy Corbyn, leader of Britain's main opposition Labour Party, called on his lawmakers on Wednesday to end "trench warfare" and unite to challenge the ruling Conservatives and deliver "socialism of the 21st century".

Corbyn, re-elected as leader at the weekend after a divisive contest, said he wanted to win back trust in Labour. The party was dealt a crushing election defeat by the ruling Conservatives last year, when many voters cast doubt on whether the party's left-wing program would protect the economy.

Despite attracting hundreds of thousands of new members to join the Labour Party, he faces a daunting task to impose his authority on his own lawmakers and sell his socialist policies to the wider electorate.

He is also facing criticism that he fails to understand Britons' widespread concerns over immigration - a key factor that prompted them to vote in June to leave the European Union.

"Our job is now to win over the unconvinced to our vision. Only that way can we secure the Labour government we need and let's be frank, no one will be convinced of a vision promoted by a divided party," Corbyn told his party's annual conference in the northern city of Liverpool.

"So I ask each and every one of you ... end the trench warfare and work together to take on the Tories," he said, using the colloquial term for the Conservative Party.

Corbyn, first elected last year on a wave of enthusiasm for a new politics, promised not to "offer false promises on immigration as the Tories have done".

That puts him at odds with some in his party, who say Labour must respond to people's worries and consider some form of control over the numbers arriving in Britain if it is to avoid alienating more voters.

"If Labour now argues for the status quo, it will look like we have abandoned them too," said Andy Burnham, the party's home affairs spokesman, who signaled he would resign from Corbyn's top team to run for mayor of the northern city of Manchester.

"So let's develop a plan for fair Brexit that deals with their concerns but supports our economy and keeps the Britain we have known - open, welcoming, playing its part in the world."

BREXIT PRIORITY

Prime Minister Theresa May has made clear that a priority in Brexit talks is to get some control over immigration, but Corbyn said a Labour government would "not sow division by fanning the flames of fear".

"We will tackle the real issues of immigration instead, whatever the eventual outcome of the Brexit negotiations and make the changes that are needed," he said.

Corbyn's calls for unity are repeated publicly by almost all in his party, but privately many centrist Labour lawmakers say they fear his win will serve to cement his authority and drive the party further left, ceding the center ground from where Tony Blair won three consecutive elections from 1997 to 2005.

Most lawmakers welcomed his speech, in which he told them to be ready for an election as early as next year, though the government's mandate runs until 2020 and May has said she has no intention of calling an early vote.

Many of those who do not see eye-to-eye with their leader say they have little choice but to play ball after Corbyn's second resounding leadership win.

But Stephen Kinnock, a lawmaker and son of a former Labour leader, described the situation as "a work in progress".

"It's mixed. I think we're almost there in some aspects of our offer, some aspects still need work," he told Reuters as a woman passing by called him a "hate-monger".

(Additional reporting by Kylie MacLellan, editing by Mark Trevelyan)