Keir Starmer has refused to back free school meals for all primary school children if Labour comes to power - saying it's too expensive.
He faces calls from within the party, with leaders in London and Wales committing to the measure.
Backers say it makes a massive difference to children in the classroom, and The Mirror is campaigning for free school meals for all primary school children.
Mr Starmer was played a clip by Labour councillor Jasmine Ali, who described it as a "no brainer".
Ms Ali, the deputy leader of Southwark Council who is responsible for education, said: "We found that pupils made four to weeks more progress than expected.
"Teachers themselves said that the behaviour in the classroom was better, pupils were more ready to learn and the concentration was there.
"If you've got hungry children, they're not going to have the concentration are they?"
But the Labour leader said he would not be pledging to bring the move in country-wide.
He told the BBC's Today programme that there was a "healthy debate" within the party, but continued: "We've said what our position is, which is breakfast clubs for every child."
Pushed on free school meals for every primary school pupil, he said: "The money is a big factor, I won't shy away from that.
"If we're privileged enough to come into power... we'll inherit a broken economy after 13 years of failure, broken public services and we have to have clear rules about what we can afford and what we can't afford."
Last week The Mirror reported that tens of thousands of voters have backed a call for Rishi Sunak to bring in free school meals.
A seven-year-old girl named Poppy wrote in a letter to the PM: “If you don’t have food, your brain won’t learn.”
Mr Sunak is urged to overturn strict eligibility rules that mean more than 800,000 kids living in poverty miss out on free lunches.
Gary Lineker, Greater Manchester Mayor Andy Burnham and London Mayor Sadiq Khan are among more than 240 politicians, groups and faith leaders who are backing the calls - which have also attracted 90,000 signatures from the public.
The Mirror and the NEU have been campaigning to extend the benefit to all primary school children regardless of background.
All children in England can get free school meals up to the end of Year 2 but after that it only applies to households on certain benefits.
Pupils living in Universal Credit families are eligible if their parents earn less than £7,400-a-year from work.
During a Q&A session after a speech in Gillingham on Thursday the Labour leader also said it was not party policy to scrap the controversial two-child benefit limit.
First introduced in 2017 it restricts benefits support to the first two children and has been repeatedly been blamed for increasing poverty and hardship.
In an interview with The Mirror last month the Shadow Work and Pensions Secretary Jonathan Ashworth described the policy as "heinous" and hinted the party would scrap it.
Mr Starmer said it was not currently Labour's policy, but added: "If it changes I'll let you know."
