He has become a familiar sight to patients and staff at Addenbrooke's, which is an internationally renowned teaching hospital and research centre in Cambridge, England.
Patients said they had been soothed by his presence and staff told of being "calmed" by his cuddles after shifts. The hospital said Henry was a "character" but urged people not to tempt him inside.
Pictures of the photogenic feline spawned a Facebook page, which sets out to " keep up morale of all at Addenbrooke's".
Henry is seven and has been visiting the Cambridge hospital for several years. Two of Jo Stranix's sisters met Henry when they came to visit their father, who has been in Addenbrooke's for several months.
They had not been able to see him due to Covid-19 visitor restrictions.
"We weren't sure if this was going to be the last time they would see him alive," she said.
The visit had been distressing but Henry was there when they left.
"They told me 'in that moment he made us all smile' and I was so thankful to that cat I'd never met for being there to comfort my sisters in some small way when I couldn't."
Henry was described as "a beautiful ginger hero" by one regular visitor to the hospital
One parent, whose 14-year-old son had been treated for leukaemia for several years, said: "Henry really does bring joy to patients and staff alike.
"You know it's going to be a good day when you get to see Henry."
A worker said Henry, who is seven, had helped her "massively during the first wave of Covid".
"Seeing Henry's ginger fluffy face at the end of the day was soothing. It's hard to explain really."
Another said: "When I've had a challenging day and I see Henry and I call his name - his tail pops up and he comes for a chin rub. Henry is a big sparkle of joy."
Henry's owners, who wished to remain anonymous, said he had a loving home nearby.
They became aware of his "exploring" via social media.
"I would try to bring him home, but that never lasted more than a couple of days," the owner said.
"It's so nice that he's making so many people happy - if he helps just one person then that's more than many cats do."
Aleks, a member of the hospital security team, has come across Henry many times as they both patrol the site.
"Only animals that support our patients are allowed inside the hospital - they have to have a harness and fluorescent strips," he said.
"But Henry does a great job just putting a smile on a grumpy person in the morning, or brings out a smile from a teenager in a wheelchair."
He said he had occasionally been called on to evict Henry from inside the buildings and added: "There would be people who are not happy that a cat is browsing around the hospital, but 99% of people love him."
"He has been a great sense of therapy and companionship to all those who are graced by his presence," she said.
"We are very lucky to have such a wonderful animal, in fact not just an animal, but a part of the [hospital trust] family."
A Cambridge University Hospitals spokesperson said: "Henry the cat is very much a character and does seem to have found a place in the hearts of many of our staff and patients.
"However, for obvious reasons of infection control, no animals other than Pets As Therapy dogs and guide dogs are allowed in the hospital.
