The firm run by Lynton Crosby, whose business partner is helping Boris Johnson run his Tory leadership bid, lobbies in Australia on behalf of a major flavoured milk brand that contains more sugar than Coca-Cola, it has emerged.

Last week Johnson announced that as prime minister he would reconsider a tax introduced last year that places a levy on high-sugar drinks. The tax does not cover sugary milk-based drinks but the health secretary, Matt Hancock, has suggested this could happen as a way of tackling childhood obesity.

Johnson’s campaign is being directed by Mark Fullbrook, who co-runs the London company in Crosby’s empire, Crosby Textor Fullbrook Partners (CTFP).

The company does not say which firms it represents in the UK.

A register of lobbying interests in New South Wales shows the Australian part of Crosby’s business, Crosby Textor Research Strategies Results, acts for the dairy multinational Parmalat.

The Italian-based company, owned by the French dairy group Lactalis, sells various brands of sugared milk-based drinks in Australia, some with higher sugar content than soft drinks such as Coca-Cola, which would subject them to an extended UK-style sugar tax.

Fullbrook worked with Johnson on his successful 2012 campaign to win a second term as London mayor. He was also behind Zac Goldmith’s losing mayoral campaign four years later, which was condemned by critics as racist for focusing extensively on the Muslim background of the eventual winner, Labour’s Sadiq Khan.

The Crosby Textor group said Fullbrook had taken holiday to work with Johnson, and had no contact with any of the company’s clients. But Jon Trickett, the shadow Cabinet Office minister, said he was concerned about the situation.

“There is a clear responsibility on MPs to avoid actual or perceived conflicts of interest,” he said. “It appears Johnson has no regard for this principle and the ministerial code would be worthless were he the person responsible for enforcing it, as the prime minister is supposed to do.

“Crosby must immediately register his firm with the lobbying register and provide a full list of the clients he represents. Johnson must also come clean as to what role Crosby had in formulating his sugar tax proposals. Both have serious questions to answer.”

A spokesman for CT Group, the parent organisation that includes the British and Australian firms, said: “As has been reported extensively, Mark Fullbrook has taken holiday to volunteer for Boris Johnson’s leadership campaign and has no contact with any clients. To suggest or imply that CT Group has anything to do with the campaign’s policy announcements is completely and utterly wrong and deliberately misleading.”

There was no comment from Johnson’s campaign team.

In January it emerged that Johnson received £23,000 in loans and donations from CTF Partners to help pay for office and staffing costs.

Crosby and his firms have been heavily involved in several Conservative election campaigns, a number of them successful. Crosby was knighted after he helped David Cameron win a majority in the 2015 election.

However, he also oversaw the disastrous 2017 Tory election campaign that cost Theresa May her majority, in which the Conservatives spent £4m with CTFP.

The Guardian subsequently revealed that Crosby’s UK company oversaw a network of secretive pro-Brexit Facebook campaigns, part of a wider effort to undermine May and push Britain towards a hard Brexit.