PORT WASHINGTON, N.Y. — When thousands of New York City nurses walked off the job last month in the city’s largest strike of its kind in decades, 9-year-old Logan Coyle was a patient in the cancer unit at NewYork-Presbyterian’s children’s hospital in Manhattan.

Logan was recovering from his latest setback in a two-year battle with advanced liver cancer that has already included chemotherapy and a complex triple transplant of a liver, pancreas and small intestine.

As the nurses formed their picket outside, he walked to his window and held up a handmade sign: “Proud of My Primaries.”

Morgan Bieler, one of Logan’s primary nurses, said the sight was a jolt of encouragement in those early, uncertain hours of the walkout, which, at the outset, involved roughly 15,000 nurses across some of the city’s most prestigious hospitals.

“In that moment, it kind of reinforced like, ‘This is why we’re doing this’,” she said recently. “If he can fight for as long as he has and as hard as he has, then we could fight this.”

But nearly a month on, more than 4,000 nurses in the NewYork-Presbyterian system are the last on the picket line in a bitter dispute over salaries, staffing, safety, health care and other contractual issues.

The hospitals have said the union’s demands were exorbitant. They say unionized nurses’ salaries already average $162,000 to $165,000 a year, not including benefits.

The nurses have countered that top hospital executives make millions of dollars a year.

Jeff Coyle, Logan’s father, says its “infuriating” that some of the city’s most vulnerable patients are caught in the middle.

“Every single day that this drags on is a severe impact to us,” he said. “We are the collateral damage of this strike.”

On Monday, the nurses’ union reached tentative deals with two other major systems, Mount Sinai and Montefiore. Those three-year proposals, if approved in membership votes this week, would see unionized nurses at those hospitals return to work by Saturday.

Negotiations at NewYorkPresbyterian, though, have progressed slower. The hospital says it has agreed to a proposal from mediators that includes many things the union has sought, including pay raises, preserving nurses’ pensions, maintaining their health benefits and increasing staffing levels.

But the union says the strike remains, and there were no plans for negotiations to resume as of Tuesday.

Coyle believes hospital administrators should have negotiated more aggressively rather than opting to hire thousands of temporary nurses to fill staffing gaps week after week.

“If we have to be there, each side to these contract negotiators should also be there, working as hard as they can to end this as quickly as they can,” Coyle said.

Spokespersons for NewYork-Presbyterian didn’t immediately comment, but the hospital systems through the strike have said they have remained ready to negotiate when called on.