* May says commitment to reform is "undimmed" after election
* Relaunch speech will mark one year in power, urge bold changes
* Tricky Brexit legislation begins later this week (Changes date, adds Green comment)
By William James
LONDON, July 10 (Reuters) - Britain's Theresa May will promise to fight on as prime minister and pursue bold domestic reform despite her diminished authority in a speech this week relaunching her leadership after 12 tumultuous months in power.
As her minority government prepares to start the difficult task of passing Brexit through parliament, May will remind Britons of her promise to build a fairer society, seeking to repair a reputation damaged by an ill-judged snap election.
Almost one year on from her first day as prime minister -- a job she took on after Britain's shock referendum vote to leave the European Union -- May will urge voters and political opponents to help her "tackle the injustice and vested interests that threaten to hold us back".
"My commitment to change in Britain is undimmed," May will say on Tuesday, according to advanced extracts from her speech.
May called the election on June 8 in an attempt to cash in on high poll ratings and win support for her plan to make a clean break with the EU in 2019.
But the gamble backfired when voters stripped her of a majority in parliament and forced her party into a deal with a small Northern Irish party to prop up a minority government.
"In this new context, it will be even more important to make the case for our policies and our values, and to win the battle of ideas both in parliament as well as in the country," May will say.
May's ability to carry on as prime minister and drive Brexit legislation through parliament with only a fragile majority behind her has been persistently questioned since the election.
But May's position seems secure for now, with colleagues preferring to see her start Brexit talks rather than topple her and risk the government collapsing -- a scenario that could let a resurgent opposition Labour Party into power.
The first stage of the Brexit process will come later this week when a key piece of legislation, which translates EU law into British law, is presented to parliament.
Damian Green, who as First Secretary of State is effectively May's deputy, told Sky News he was confident the government could get the legislation through .
Asked about media reports of plots to oust her, Green said it was simply "gossip and chatter" at summer parties.
"There is no credible plot going on," he said. "The prime minister is determined to carry on, to lead the party and the country for many years to come and the overwhelming majority of Conservative MPs (lawmakers) are behind her in that."
May will make the appeal for national backing at the launch of a report into how the government should protect workers affected by a shift to different employment models, including the 'gig economy' championed by the likes of Uber and Deliveroo.
The report is a key element of May's plan to address dissatisfaction expressed at the referendum by working class Britons who felt left behind by globalisation.
"At this critical time in our history, we can either be timid or we can be bold," May will say. "We will act with an unshakeable sense of purpose to build the better, fairer Britain which we all want to see." (Additional reporting by Kylie MacLellan, Editing by Catherine Evans)