The Manchester United and England star has kicked off his campaign by saying ‘we must act with urgency’ to make sure no child worries about what they are going to eat.
Currently, 217 people in Chesterfield have so far signed the petition urging the Government to take action.
It calls for free school meals to be extended to every child from a household on universal credit or an equivalent benefit.
A Government-commissioned report published earlier this year said expanding the scheme in this way could reach an additional 1.5 million seven to 16-year-olds across England.
The petition also calls for free meals to be provided throughout holidays, as well as during term time.
Finally, it says healthy start vouchers – given to eligible women who are pregnant or those with young children to buy basic foods – should be raised from £3.10 to £4.25 per week and made available to all those on universal credit or a similar benefit.
Across the UK, tens of thousands of people have already signed the petition since it went live on October 15.
The campaign is UK-wide but decisions around school meals are devolved.
Marcus Rashford forced the Government into a U-turn over holiday food vouchers during the pandemic earlier this year.
He said: “Child food poverty in the UK is not a result of Covid-19, we must act with urgency to stabilise the households of our vulnerable children
“In 2020, no child in the UK should be going to bed hungry, nor should they be sat in classrooms concerned about how their younger siblings are going to eat that day, or how they are going to access food come the holidays.”
The Government appeared to reject the calls for further help, with a spokesman saying: "We are in a different position now with schools back open to all pupils."
More than 5,000 children across Chesterfield were eligible for free schools meals in the last school year, the latest Department for Education figures show.
Across England, around 1.4 million children had the right to claim.
But the Food Foundation think tank, which is working on the campaign, estimated that nearly one million additional children have recently been registered for the scheme as Covid-19 drives more families into poverty.
Anna Taylor, the charity’s executive director, said: “While children have been spared the virus, they have not been spared it’s economic impacts and we must act to protect them.”
A government spokeswoman said: “We have taken substantial action to make sure children and families do not go hungry by extending free school meals when schools closed, increasing welfare support by £9.3 billion, and giving councils £63 million to provide emergency support to families for food and other essentials.
“We also appreciate it has been a challenging time for families, which is why we have increased the safety net available to them with income protection schemes, mortgage holidays and support for renters.
“We are grateful to Marcus Rashford for his work shining a light on the significant challenges families are facing at this time.”