David Davis, the Brexit secretary, will present MPs with a formal policy paper setting out how the UK proposes to leave the EU, a day after MPs voted in favour of giving Theresa May the power to trigger article 50.

He is set to address the House of Commons shortly after midday on Thursday, with the document published at the same time.

May had initially resisted publishing a white paper, suggesting that her Brexit speech contained enough detail, but relented under pressure from Conservative backbenchers and opposition parties.

It comes as John McDonnell, the shadow chancellor, promised that Labour would fight against May’s pursuit of a “reckless Brexit” after the party was whipped in favour of giving her the power to trigger article 50.

Jeremy Corbyn’s closest ally said Labour would not give the prime minister a “blank cheque for what others call a hard Brexit and I think is a reckless Brexit”.

“We are all uncomfortable with the position, we campaigned to remain, but we are democrats and have to respect the will of the people,” he told BBC Radio 4’s Today programme.

McDonnell said Labour would come together as a party to shape Brexit, after 47 MPs defied the party whip to vote against triggering article 50, which begins the formal EU exit process.

He also hinted that the party may not sack more than a dozen junior frontbenchers and whips who voted against Brexit, even though three shadow cabinet ministers resigned.

McDonnell said Labour had not decided how to whip the final vote on article 50, but the party would not block Brexit, raising the possibility of abstentions.

Diane Abbott, the shadow home secretary, who avoided the vote on Wednesday night because she was ill, has previously said the position would be reviewed before a third reading in the Commons.

Even the former chancellor George Osborne acknowledged that the battle over the direction of Brexit would begin after the parliamentary vote, McDonnell said.

“What we are going to fight for in the coming period is to regain the future based on a thriving economy, jobs protected, a constructive relationship with Europe and the rest of the world, the protections of workers’ rights, environmental regulations, consumer regulations; all those benefits we got from the EU we want to preserve, but we want to tackle some of the perceived disbenefits that motivated people to vote leave,” he said.

The shadow chancellor also highlighted the rebellion among the pro-EU Liberal Democrats. Two out of the party’s nine MPs abstained, while the rest voted against the bill.

No 10 will be hoping the white paper satisfies would-be Tory rebels who could threaten to back opposition amendments if they feel there is not enough parliamentary scrutiny of the process.

The legislation is still highly likely to pass through parliament and make it into law, allowing May to meet her self-imposed deadline of triggering article 50 before the end of March.