Children’s Commissioner Dame Rachel de Souza said a fifth of children in England also felt this way – making it their top concern – and praised them as being part of a “survivor generation”.
She polled more than half a million school pupils across the country for her Big Ask survey between April and May.
In Peterborough, 797 children aged between nine and 17 responded to a question on their mental health – with 20% saying they were unhappy with it.
Meanwhile, 12% of kids said they were unhappy with their physical health, 5% with their friendships, and 8% with their life overall.
The survey found that a fifth of children across England were unhappy with their mental health – but girls were almost twice as likely as boys to think this (25% versus 13%).
Dame Rachel said this generation were not ‘snowflakes’, but were “veterans of a global crisis”.
She added: “They have seen how colossally frightening life can be, far too young, and have made a lot of sacrifices.
“But they have endured and are emerging stronger and prematurely wise. Bruised, yes, and in many cases seriously vulnerable, but, for the most part, happy, optimistic and determined.
“They are a survivor generation – a sleeves‑up, pragmatic generation, with civic‑minded aspirations.”
The report is calling for a comprehensive catch-up package for schools, a faster expansion of mental health support teams, and stronger safeguards for social media platforms.
The survey also asked pupils what they worry about, with the highest proportion nationally (41%) saying they were concerned about having enough money to buy the things they need.
The second most common worry reported (39%) was whether they will grow up to benefit from a healthy planet.
In Peterborough, an equal proportion of children (38%) said they were worried about having a good job or career, having a healthy environment and planet, and having enough money to buy the things they need.
On the whole, pupils in the area are more optimistic than kids elsewhere.
Around 56% said it is likely they will have a better life than their parents, compared to 52% across England.
Catherine Roche, chief executive at mental health charity Place2Be, said the coronavirus pandemic has exacerbated existing inequalities.
She added: “Our school-based mental health professionals have seen an increase in concerns around self-harm and suicidal thoughts in our secondary schools.
“However, we also know that there was already a big gap in support for children and young people even before the pandemic.”
Education Secretary Nadhim Zahawi said the Government has already taken action to address young people’s concerns, including an Online Safety Bill, committing to Net Zero and hosting COP26 later this year.
He added: “We know that the pandemic hit young people hard, which is why we have launched a tutoring revolution to make sure they catch up and bolstered mental health support in schools.
“As we drive to level up opportunities across the country, we will continue prioritising young people’s wellbeing alongside academic success.”