Andy Burnham is being urged to protect 4 million gig economy workers by prosecuting companies that use widespread “bogus” self-employment practices to deny people their rights.

As the prime minister fleshes out his agenda, the Fabian Society and the Joseph Rowntree Foundation have warned that millions of workers are missing out on key statutory rights including parental leave, redundancy pay and protection against unfair dismissal.

Publishing a joint report calling on Burnham to take action, the Labour-linked thinktank and the anti-poverty charity said changes were essential to make work pay amid the cost of living crisis.

The report calls on the government’s Fair Work Agency – launched in April as a powerful watchdog for workers’ rights – to use its civil proceedings powers to prosecute companies suspected of engaging in “widespread bogus self-employment”.

It also urges ministers to push ahead with changes to employment law to combat longstanding misuse of the system.

Britain’s employment laws are unusual among leading economies for including three classifications of employment status. An employee has the most rights and responsibilities, while a self-employed person has limited protections but fewer obligations. There is also an intermediate “worker”, or “limb (b)” status, with fewer rights and obligations than a full employee.

According to the report, as many as 4 million workers – including hairdressers, personal trainers, delivery workers and drivers for gig economy platforms – work in this gap. Experts say many have been hired by gig economy companies in this way to avoid granting them employment rights.

Labour had promised to create a single “worker” status for all but the genuinely self-employed, before rowing back on the plan to fend off Conservative attacks on its approach to business before the 2024 general election.

Once in power, the policy was not included in Labour’s shake-up of employment rights, and though it promised to consult on “moving towards a single status of worker”, this has not materialised.

Labour faces intense lobbying from business over its sweeping Employment Rights Act, which includes a ban on exploitative zero-hours contracts and day-one rights to statutory sick pay.

Last week, official government analysis showed the planned changes would help to support economic growth and offer workers more protections, though they would cost businesses an estimated £350m to £2.9bn.

In his first week in office, Burnham spoke to business leaders to state his determination to push ahead with the plan for the biggest shake-up of employment rights in a generation.

However, the Fabian Society and Joseph Rowntree Foundation said changes to address employment status were needed to ensure gig economy workers would benefit from these upgraded protections.

The report calls for the burden of proof for whether someone is an employee, worker or self-employed to be shifted from the worker to the employer. This would help people to avoid fighting lengthy legal battles to secure basic rights such as the minimum wage and holiday pay.

Luke Raikes, the deputy general secretary of the Fabian Society, said: “The government must ensure those who play fast and loose with employment law can’t undercut the majority of good employers who play by the rules.

“If it looks like employment then employers should be treating people as employees by default and should expect enforcement action if they aren’t complying. Insecure workers should not have to take on a company’s lawyers – sometimes all the way to the supreme court – just to establish the basic rights they are owed.”

A government spokesperson said: “Our plan to make work pay will boost secure employment, improve job security and deliver stronger, higher living standards. The Fair Work Agency is already helping workers get the rights they are entitled to. We’ve committed to consult on employment status and will consider how the Fair Work Agency can use its civil proceedings power alongside that.”