The England footballer after his campaign to extend the food voucher scheme into the summer holidays, despite a spokesperson for the Department for Education initially stating the national voucher scheme “will not run during the summer holidays”.
Heath secretary Mr Hancock told ITV News Rashford's intervention was "extremely powerful" and "very moving" - but suggested it facilitated a change that may have already been coming.
"Initially it was a yes to an extra £60 million going through local authorities and then we kept looking at it and then the prime minister made his judgement," he told Political Reporter Shehab Khan.
"The funny thing is that if the government never changes its view then you'd say that it didn't listen...any new policy development involves changing things and changing things isn't necessarily a bad thing."
Mr Hancock even jokingly said Premier League footballers should join the Cabinet: "Get elected, join public life, it's hard but the service is worth it."
The health secretary later blamed the early morning start for using the wrong name and said he may have had "Harry Potter on the mind" when he referred to Marcus Rashford as "Daniel Rashford".
"My seven-year-old listens to Harry Potter and reads Harry Potter avidly, including at 5.30 this morning when I got up to do this morning’s media round," he told LBC.
Responding on Twitter Rashford wrote: "I’ve been called much worse over the last couple of days."
The 22-year-old has now said he is considering the "next steps" for the campaign.
He told BBC Breakfast "I think this is only going to be successful throughout the summer period so we’ve bought ourselves six weeks of time now to plan and figure out what’s next and keep taking steps forward.
"I don’t want this to be the end of it because there are definitely more steps that need to be taken. So we just need to analyse the response."
The Manchester United striker added that people are "struggling all year round" and said he now wants to learn "how we can help them best" in the future.
Rashford initially wrote an open letter on Monday asking the government to reverse its decision not to award free school meals vouchers to vulnerable children outside term time.
In his initial call for the U-turn, Rashford spoke about his personal experience relying on similar schemes:
But, amid mounting pressure, Number 10 said on Tuesday that the government would provide a Covid Summer Food Fund to reflect the fact that families face an "unprecedented situation" over the holidays.
Downing Street announced a one-off £120 million fund - which will benefit some 1.3 million children in England over the six-week summer period.
Sir Keir added the PM is an "expert" in U-turns with three in the space of a month.
The PM told the Downing Street press conference: "Clearly free school meals should generally apply in term time, that’s what they are there for.
"But we have to understand the pressures families are under right now and that’s why we’ve responded as we have.
"As I say, I think it is the right thing to do and it will help the kids from the families who really need it."
It is understood that payment will be in the form of a one-off six-week voucher given to eligible families at the end of term to use in supermarkets.
The vouchers will amount to £15 a week per child and some 1.3 million children in England will be eligible, the Prime Minister’s spokesman said.