Government ministers have been urged to set targets to boost the number of male teachers in England, as backbench MPs seek to capitalise on turbulence in the Labour party to influence government policy.

With the government in disarray after the shock resignation of the defence secretary this week, MPs are seizing the moment to embark on a battle of ideas, including tackling toxic masculinity, which they argue has played a role in violent anti-immigrant disturbances.

The Labour Men and Boys group are also putting pressure on Keir Starmer, or his replacement, to increase paternity leave. More than 50 MPs, including the former deputy leader Angela Rayner and the former health secretary Wes Streeting, have worn England and Scotland football shirts calling for more than two weeks of paternity leave, which is among the worst in the OECD group of rich countries.

The MP for Hitchin, Ali Strathern, said that while the Westminster summer had been dominated “by personality stories and drama”, voters wanted a clear message that the government was on their side.

“For the rest of the country, this summer is about football, family and the cost of living,” said Strathern, the Labour group’s co-chair. “That’s why over 50 government MPs have donned these shirts calling for extra time for dads – because the UK’s paternity leave offer is the worst in Europe and two weeks isn’t enough.”

He said the government’s pledge to recruit 6,500 additional teachers by the end of the parliament was welcome and should be accompanied by national marketing campaigns similar to those that have encouraged girls and women to seek jobs in Stem.

After unrest in Belfast and Southampton, it was vital that the government also recognised that boosting the number of male teachers was a key tool in combating the manosphere, said Peter Swallow, a former teacher and the MP for Bracknell. Only 24% of teachers in England are men, according to DfE data. In primary schools the figure is 14%; the figure is 35% in secondary education and 3% in early years.

“There is a crisis of masculinity in this country and boys who are feeling vulnerable, not listened to and isolated are too often turning to the easy answers offered to them from the manosphere, who want to sell them on a very narrow idea of what it is to be a successful man,” he said. “Getting more male teachers and more positive role models in their lives has to be part of the solution.”

With all eyes on the Makerfield byelection on Thursday, MPs see this week as a critical moment for their ideas ahead of a potential Labour leadership campaign, in which the Greater Manchester mayor, Andy Burnham, could challenge Starmer.

“It feels like all ideas are on the table again,” said one MP. “When you think about the legacy of this Labour government, whether this government likes it or not in 20 years people are not going to be speaking about breakfast clubs – but they will remember if we significantly boosted paternity leave and gave families time together in those critical early days.”

On Monday, dozens of MPs will wear custom shirts featuring the phrase “Extra time for dads” on the front, with “2 weeks isn’t enough” on the back. The stunt, organised in conjunction with Movember and the Dad Shift, will also target football stadiums with a World Cup-themed paternity leave poster campaign.

Research from the pro-dad groups found that 92% of the public agree it is important or very important for fathers to have time with their babies in the first weeks and months, while 60% back increasing paternity leave length and pay.

A Department for Education spokesperson said it was 70% of the way to recruiting 6,500 new teachers and was offering tax-free bursaries and scholarships worth up to £31,000. “We know there are too few men working in our schools. While a lack of male teachers is a historic and worldwide problem, we are looking at a range of options to improve diversity in the workforce, including attracting more men to the profession,” they said.