Train chiefs had promised a structure would be put up on Platforms 2 and 3 at the station so passengers weren't left exposed to the elements during the coldest months of the year.
The platforms have been without shelter since a waiting room was demolished in 2019 and plans by another operator to build a replacement, with toilets, stalled due to Covid.
Pete Myers, Northern's stakeholder manager, told a public meeting on Thursday that building a temporary shelter had proved impossible because it would not have withstood the weather.
He said: "I do apologise to our customers who've had to stand outside this winter. It's certainly not something we'd have wanted them to do.
"We did look at putting a temporary structure up but in the end the wind was so great it would have been ludicrously costly.
"It's on an embankment and it's so open there it would have probably ended up at Wakefield Westgate."
Mr Myers insisted that the plans to build a new permanent waiting room would still go ahead and would be completed by Grand Central.
However, he said the operator had been hit so hard by Covid they'd barely "run a train" throughout the pandemic.
He added: "The issue with this is actually getting the concrete over to that part of the station to put the foundations in.
"There were several stalled attempts to do that last year.
"We helped as best we could, but each time it cost Grand Central a great deal of money to do it.
"Frankly when everything shut down and they furloughed staff, they pulled away from the project. "It is back on now and it will happen.
"They will get loos, they will get a heated waiting room and everything that was promised and to be fair to Grand Central they are very keen to get it done, but when they're closed they're closed and there's no-one to speak to about it."
Northern said it's passenger numbers across its network are currently between 15 and 20 per cent of pre-Covid levels.
Although some services were cut in response to the third lockdown being announced in January, Mr Myers most trains would be restored on May 16.