The Mid Yorkshire NHS Hospitals Trust, which runs Pinderfields, Pontefract and Dewsbury Hospitals, has consistently missed its cancer target during 2018.
The government says that 85 per cent of patients with the disease should get their first treatment within 62 days of being referred by their GP.
The snow and over-reliance on individual departments have been blamed for contributing to the problem at different times this year.
But now staff say they are being overwhelmed by the number of referrals for those with prostate cancer symptoms, after a publicity drive by famous figures encouraging men to visit their GP if they are concerned about it.
Trudie Davies, the trust's director of operations, said that the issue also related to a lack of space and MRI equipment.
She said: "At the moment it's a real challenge. There have been a lot of high profile people in the press who have raised the issue of prostate cancer, and we've been hit by a massive increase in volume because of the awareness.
"The increase in awareness is great, but we've not been able to step up.
"It's not really a finance issue. Extra money might have helped stem the flow a bit but it's more of a capacity issue."
Former BBC Breakfast presenter Bill Turnbull is among those highlighting the illness.
He said earlier this week that "unbearable" chemotherapy had stopped his prostate cancer spreading, but it hadn't "been beaten back entirely".
Trusts across the country are struggling to hit the 62 day target, to the point where patients are no longer being passed between areas.
Mrs Davies added: "Really this is a capacity problem that's hit the whole of West Yorkshire. There is an underlying problem within cancer care. No-one can help each other. No one trust is doing better than anyone else.
"If we put one week's worth of demand on a smaller trust that would wipe them out for four months. Equally if Leeds sent their patients to us that would wipe us out for four months.