The Labour Mayor for Greater Manchester texts Louise Haigh each Monday urging her to take action against Northern Trains, which was nationalised in 2020.
Ms Haigh has vowed to improve services "as a priority".
Northern runs services to cities including Manchester, Liverpool, Newcastle and Sunderland, as well as to parts of the Midlands, but performance has plummeted in recent years with just six out of ten trains arriving on time.
The number of cancelled trains rose from 1.35 per cent of the total in 2020-21 to 5.3 per cent in 2023-24, meaning more than one in 20 services on the timetable simply don't run.
Drivers and conductors on parts of the network are not contracted to work Sundays, with the service relying on overtime volunteers.
The operator has asked bus firms to provide replacement services over Christmas, and admits it still uses fax machines to communicate with staff.
Mr Burnham has accused the operator of "ruining people's lives".
Northern is run by the Operator of Last Resort, a business owned by the Department for Transport. Under the Government's nationalisation plans the OLR will be handed control of more franchises until a new rail operator, Great British Railways, is created to oversee the network.
Ms Haigh promised to ensure Northern staff are contracted to work on Sundays, which is expected to reduce the number of cancellations.
She said: "We need to get drivers and conductors on modern terms and conditions that reflect the railway that we need."
And, she added: "I was struck when I met Network Rail and managing directors of the worst-performing operating companies that they made it clear the previous levels of performance had simply been tolerated and accepted as normal."
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