MPs have called on the government to accelerate a proposed takeover of Southern metropolitan services by Transport for London, as the train operating company Govia Thameslink Railway prepares to cut back its timetable by 350 trains a day.

An early-day motion sponsored by Labour former shadow business secretary Chuka Umunna, and backed by MPs of other parties in affected constituencies, has added to opposition calls to strip GTR of its franchise. It proposed that TfL run the trains instead by adding them to its London Overground network.

The government signalled last year that it would look to bring more lines under TfL control, but that this would not happen until at least 2021, when the franchise expires.

Umunna, the MP for Streatham, has also launched a public petition to demand GTR be stripped of the franchise, citing hundreds of complaints from constituents and commuters on the trains serving stations in his constituency. He said they included students who had missed exams and people who had lost jobs.

Umunna said: “It goes from bad to worse. Every time we’re promised improvements, it goes backwards. It’s simply not acceptable that they think they can get around the problems by cancelling the trains.”

On Tuesday, GTR confirmed it would be cancelling 341 trains daily from next Monday in an attempt to end “random cancellations” and provide a more reliable service to passengers. Chief operating officer, Dyan Crowther, told the Guardian on Thursday that the timetable would run “until further notice”.

She said the company hoped goodwill gestures would help tackle staff shortages in the short term and could end the breakdown of relations with the railway workers’ union, RMT, opening up the possibility of fresh talks.

“We’re trying to get our staff back to work,” she said. “We can’t force people to work overtime, because it’s a voluntary activity ... But what we can do is talk to our staff about what their concerns are – restoring travel passes and car parking will be seen as a positive gesture.”

She added: “What we’ve experienced in the last two months is a very difficult situation around industrial relations with unions who do not want to support us in modernising the railway. This is about delivering better services, and what we’ve got is two trade unions standing in the way.”

However, Umunna said the problems predated the industrial dispute, which concerns changes to the role of conductors and led to strikes earlier this year. He said: “The real problem is poor leadership and poor foresight by the company. GTR have proved themselves to be completely incapable ... The real problem is this franchise is too big.”

The new franchise, which incorporates Thameslink, Great Northern and Southern services, including the Gatwick Express, now accounts for about one in four journeys in the UK.

After raising the question in the Commons, Chris Grayling, the leader of the house, said he had “every sympathy”, adding: “What is happening right now is simply unacceptable, and it has to be sorted.”

So far, ministers at the Department for Transport have ruled out moving against GTR. A Westminster Hall debate on the future of the franchise will be held next week.