The Mid Yorkshire Hospitals NHS Trust said a total of 187 patients have died with the virus after they were thought to have contracted it while in the trust's care.
The trust runs Pinderfields Hospital in Wakefield, as well as Pontefract and Dewsbury Hospitals.
However, it is not yet known for definite how many of the 187 died because of Covid, as opposed to with Covid.
The number itself is also uncertain, because of the time lag between catching the virus and showing symptoms, but all of those concerned were first admitted to hospital for other reasons and later tested positive after their eighth day in hospital.
The trust, which released the figures at a public board meeting on Thursday, said there was as yet no evidence of systemic negligence on its behalf.
A small number of compensation enquiries have been forwarded to the NHS' litigation branch, although it's understood these have not yet materialised into formal claims.
Speaking at the board meeting, trust chief executive Martin Barkley said: "This is a really sad and sensitive matter.
"Despite our very best endeavours, we've not been able to prevent patients contracting Covid while they were with us.
"We are one of many trusts that have had this difficulty.
"We recognise that this has been a very sad time for patients who've become ill with Covid because of hospital-acquired infection and especially for those families who've been bereaved as a consequence of their loved ones dying from Covid."
The trust said that around 94,000 patient admissions were made across its three hospitals between April 2020 and March 2021, including just over 4,000 Covid cases.
In total, 724 patients tested positive for coronavirus after their eighth day in hospital, having been first admitted with other ailments.
Mr Barkley said one factor behind the figures had been the number of Covid admissions the trust has had to deal with.
According to healthcare info unit Dr Foster, only 22 hospitals across the country treated more Covid patients than Mid Yorkshire during the pandemic's first 12 months.
Asked if there was any evidence of negligence on behalf of the trust, Mr Barkley replied, "The answer is no, so far.
"Infection prevention control teams investigate any outbreak which occurs in real time."
The trust chief explained that advice had been sought from NHS England during an intense period of outbreaks.
NHS England could find no fault with the hospitals' processes, although they advised medics to test patients on their third day in hospital, as well as on the first and fifth days as the trust was already doing.
Although healthcare sites across the country have had similar troubles with infection control during the pandemic, the government has not insisted trusts release details on how many patients may have caught Covid on site.
Some across the country have already done so, but Mid Yorkshire is the first trust in West Yorkshire to pro-actively put its figures into the public domain.
Senior trust sources have said they want to be open and transparent with the public by releasing the data.
Mr Barkley said the trust had set up a dedicated team to work through the duty of candour process for each of the 187 patients, which involves liaising with families and offering a full explanation as to why their loved ones died.
That work is expected to take several months.
Local Democracy Reporting Service