MOSCOW — Assailed by critics as an absentee leader at the start of the coronavirus crisis in Russia, President Vladimir V. Putin re-emerged with a splash on state television last month to show that he cared and was taking charge.

He promised cash bonuses of up to $1,100 a month for each doctor, nurse and other “front line” health worker involved in fighting the virus.

But for an all-powerful leader whose every word must be taken as a command, Mr. Putin has had a surprisingly hard time making his voice heard.

More than a month after he spoke, the money has yet to materialize for many. Instead, some doctors have received visits from police investigators and prosecutors demanding to know why they complained publicly about not getting their bonuses.

A promise meant to showcase Mr. Putin’s proudest achievement — the revitalization of the Russian state after the chaos of the 1990s — has sunk into a swamp of recrimination, security service intimidation and bureaucratic buck-passing.

“Is this a joke? Unfortunately, no,” Dmitri Drize, a Moscow-based editor, wrote last week in a scathing newspaper commentary on the unfolding mess.

He said that neither Russia’s foreign foes nor its main opposition leader, Aleksei A. Navalny, “is capable of damaging the image of the state as much as its own managers.”

The Kremlin holds more than $500 billion in various rainy day funds, so Mr. Putin has all the money he needs to deliver on his promises. But, in a system rife with corruption, many officials live in permanent fear of being criticized, or worse investigated, for spending state money not included in their previously approved budgets.

So when it came to doling out the cash, they hesitated, took the liberty of making deductions for time health workers spent on non-coronavirus patients or perhaps skimmed some of the money.

In the southern region of Krasnodar, a widely respected head doctor at a hospital was fired after his staff staged a small protest. He is now under investigation by Russia’s equivalent of the F.B.I. for criminal negligence.

A doctor in the nearby town of Abinsk who helped organize public complaints over nonpayment of Mr. Putin’s bonus received a letter from the police warning that he faced prosecution for “carrying out extremist activities.”

Yulia Volkova, a Krasnodar doctor who leads the local branch of Doctors’ Alliance, an independent trade union affiliated with Mr. Navalny, said in a telephone interview that medical workers had rejoiced at Mr. Putin’s promise of extra cash. Now, though, they are “terrified of being investigated” if they complain about the president’s orders’ falling on deaf ears, she said.

In some cases, however, prosecutors have sided with protesting doctors. The prosecutor’s office in Simferopol, the capital of Crimea, annexed by Russia in 2014, said last week that it had investigated complaints of nonpayment and found them justified. It ordered local authorities to pay up.

In Nizhny Novgorod, another region where many medical staff have not received the money promised, the regional health minister, David Melik-Husyenov, accused the opposition of using “dirty tricks” to expose the bureaucracy’s failures. “Arranging such stories is very immoral,” he said.

Mr. Putin, playing one of his favorite roles as a caring but stern father of the nation undermined by bungling bureaucrats, fumed recently in a teleconference that officials in many places had not acted on his bonus order.

“I gave specific figures for these payments for doctors, for nursing staff, for all medical staff, for ambulance crews and so on,” Mr. Putin said. Instead, he continued: “They made a bureaucratic mess, counting the number of hours worked on some kind of clock. Did I instruct that you count with a watch or something? No!”

He said earlier that 29 regions had ignored his order and that less than half of medical workers nationwide had received the money he had promised. Ordering officials to get with his program, Mr. Putin thundered, “I ask you to keep in mind that I will personally check the situation on this issue in every region of Russia.”

That things have gone so awry is a measure of the wide gap between image and reality in a country that revolves around what Mr. Putin calls the “power vertical.” This is the rigidly top-down — and, in theory, stringently efficient — system that he has spent 20 years building to replace the decrepit state structure he inherited from his predecessor, President Boris N. Yeltsin.

Tatiana Stanovaya, an expert on Russian politics, said the “power vertical” has always been a political project focused on protecting the Kremlin from opponents, not on delivering efficient administration for the public’s benefit.

“It has never been effective in routine management. This is not something Putin knows how or wants to do,” she said.

“Nobody deliberately defies Putin or lets him down,” she added in a telephone interview. ‘‘That is impossible. But nearly everyone does it unintentionally because they are afraid of taking decisions.”

Much of the blame for unpaid bonuses has now fallen on the staff of Prime Minister Mikhail V. Mishustin, who spent much of this month in the hospital recovering from Covid-19.

The prime minister complained in a conference call with officials shown on television that documents needed to turn Mr. Putin’s promise into action had not been drafted properly and left too much room for regional officials to wriggle out of paying.

One of the first signs that Mr. Putin’s bonus program was going off the rails came in early May when ambulance drivers, paramedics and others gathered outside the main hospital in Armavir, a town in the southern region of Krasnodar. “We have received nothing. Not a ruble, not a kopeck,” they chanted.

A video of their protest appeared online, stirring copycat actions across the country. Krasnodar’s governor, Veniamin Kondratyev, responded that he was aware of “many complaints” of nonpayment and vowed to “investigate the situation in detail” to make sure Mr. Putin’s promise was fulfilled.

An official commission was then sent to investigate the Armavir hospital and quickly found a scapegoat: the head doctor, Sergei Smirnov. Accused of not filling in the necessary paperwork on time, he was branded as the main culprit in media outlets controlled by the regional authorities.

Reports last week that Dr. Smirnov had been fired provoked only more protests by exhausted and irate medical staff.

A group of nurses gathered near the Armavir hospital to sing Dr. Smirnov’s praises, saying that he had worked hard to make sure his staff had proper protective equipment, and to warn, in the words of one angry nurse, that “without him work will stop.”

Vladimir Lotnik, a resident who signed a petition protesting Dr. Smirnov’s dismissal, said in a telephone interview that officials were scrambling to protect themselves up and down the system by blaming the powerless. “A fish rots from the head,” he said.

Nikolai Petrov, a political scientist, dismissed Mr. Putin’s public anger and dismay over the bonuses as mostly theater.

“He is trying to show that he is the good guy,” Mr. Petrov said. “But he is losing popularity and will continue to lose it.”

An opinion poll by the Levada Center, an independent polling organization in Moscow, found that the president’s approval rating sank last month to 59 percent, its lowest level since he came to power in 2000. His highest approval rating, nearly 90 percent, came after Russia seized Crimea from Ukraine in 2014.

After nearly two months cooped up in his country residence outside Moscow, Mr. Putin has become so isolated, in Mr. Petrov’s view, that he “risks returning to a changed country after the pandemic is over.”

He likened the situation to what Mikhail S. Gorbachev, the last leader of the Soviet Union, faced when he returned to Moscow after the collapse of a coup attempt in 1991 that had kept him isolated for days in a villa in Crimea.

Mr. Gorbachev, his authority drained, resigned soon afterward.

Predictions of Mr. Putin’s imminent eclipse, however, have invariably proved wrong, and many expect the president to bounce back from these travails, albeit in a weakened position.

Russia, with more than 350,000 reported coronavirus cases, is the third most infected country after the United States and Brazil. Kremlin-controlled media outlets, however, have presented Russia’s response to the pandemic as a triumph, trumpeting official figures that show a remarkably low death rate of 2 per 100,000 people, compared with 30 in the U.S. and 55 in Britain. This, they say, is a “Russian miracle.”

The pandemic has nonetheless disrupted the centerpiece of the Kremlin’s political program for the year, forcing the cancellation of an April referendum on constitutional changes that would allow Mr. Putin to brush aside term limits and stay in power until 2036.

But, with the recent lifting of a nationwide lockdown order — despite a steady rise in the number of infections — the Kremlin is expected to push ahead with its vote on the Constitution as early as June.

The referendum’s outcome, like nearly all votes in Russia, is in little doubt, and it would secure Mr. Putin’s unassailable position for many years to come.

Opening the door for him to stay in power indefinitely, Kremlin critics say, would only entrench the dysfunctions of a system that for 20 years has paid lip service to the stated goals of a single man but often confounded them instead.

“The diagnosis is obvious,” said Mr. Drize, the editor. “Officials have forgotten how to make decisions on their own. And this disease is worse than the coronavirus.”

Oleg Matsnev and Sophia Kishkovsky contributed reporting.