At the Veterans Affairs hospital in Pittsburgh, researchers spent months preparing for a clinical trial of a new drug to treat advanced cancers of the mouth, throat and voice box.

They were ready last month to start enrolling patients — veterans whose cancer had spread to other tissue and who had run out of treatment options.

Then a problem arose.

The hospital was unable to renew the job of a key staff member involved in running the study, a typically routine process thwarted by a hiring freeze imposed under the government-cutting project led by President Trump and Elon Musk. Suddenly, the clinical trial was on hold.

“They were ready to enroll,” said Alanna Caffas, the chief executive of the Veterans Health Foundation, which administers the trials. “They had the lab kits on site. They had the drug to dispense. But they couldn’t get the clinical research coordinator renewed.”

While Trump administration officials have promised to preserve core patient services, initial cuts at the V.A. have nonetheless spawned chaotic ripple effects. They have disrupted studies involving patients awaiting experimental treatments, forced some facilities to fire support staff and created uncertainty amid the mass cancellation, and partial reinstatement, of hundreds of contracts targeted by Mr. Musk’s Department of Government Efficiency.