Russian forces made significant gains in Ukraine on Saturday, advancing into the besieged port of Mariupol, destroying an underground weapons depot in the west and leaving a marine barracks in ruins following one of the deadliest rocket strikes on Ukraine’s military in the nearly month-old war.

As the fighting raged, Ukraine faced a worsening humanitarian crisis, and military losses mounted on both sides. A senior Ukrainian military official said on Saturday that the strike on the barracks, which happened Friday in the southern city of Mykolaiv, had killed more than 40 marines.

At the city morgue, the bodies of dozens of marines in uniform were laid out side by side in a storage area. A morgue employee would not specify the number of dead brought from the site of the attack.

“Many,” the employee said. “I won’t say how many. But many.”

Separately, Russia’s Defense Ministry claimed on Saturday that it had used a hypersonic missile for the first time to destroy an underground ammunition depot in the western region of Ivano-Frankivsk. Ukraine’s military spokesman confirmed the hit on Saturday, but said the missile type was “yet to be determined.”

If confirmed, the Russian military’s use of a new generation of its missiles, called Kinzhal or Dagger, would mark an escalation in the conflict. Hypersonic missiles are capable of flying at five times the speed of sound, according to military analysts.

President Volodymyr Zelensky of Ukraine on Saturday appealed directly to Russians to support a diplomatic resolution to the war, and added a stark warning.

“I want everyone to hear me now, especially in Moscow,” Mr. Zelensky said in a video address hours after President Vladimir V. Putin of Russia had spoken to tens of thousands of Russians in a rally at Moscow’s largest stadium. “It’s time to meet, time to talk. It is time to restore territorial integrity and justice for Ukraine. Otherwise, Russia’s losses will be so huge that several generations will not be enough to rebound.”

He repeated the Ukrainian military’s claim that 14,000 Russian troops had been killed. The Pentagon estimates that the figure is about half that, still a staggering toll, one that U.S. officials say the Kremlin has sought to cover up.

“Just imagine, 14,000 corpses and tens of thousands of wounded and maimed people at that stadium in Moscow,” Mr. Zelensky said. “There are already so many Russian losses as a result of this invasion. This is the price of war. In a little more than three weeks. The war must end.”

Street battles broke out in Mariupol, the southeast port city that the Russians have besieged since the conflict’s early days, as Russian troops and allied irregular forces moved into the city after weeks of devastating missile barrages turned it into a wasteland of bombed-out buildings.

Residents who fled the city in recent days have described scenes of dead bodies dotting the streets, widespread looting and the suffering endured by thousands who remain trapped without heat or water.

Corpses “were lying in the street, just covered in rags,” said Eduard Zarubin, a Mariupol doctor who escaped the city on Wednesday. “No one cleaned up because there were no ambulances, or the ones that were still working had too much to do.”

A video shared by the pro-Russian leader of the Russian region of Chechnya, Ramzan Kadyrov, purported to show Chechen fighters, known for their merciless war tactics, inside Mariupol.

The New York Times has not independently verified the video’s contents.

Mr. Kadyrov has claimed that thousands of fighters loyal to him are now fighting for Russia in Ukraine. However, some Chechen rebel leaders who fought for the region’s independence against Russia in the 1990s have promised to join the fight on Ukraine’s side.

Russia’s Defense Ministry said in a statement that it was “tightening the noose” around Mariupol. And the Ukrainian government reported that its forces were outgunned, that attempts to provide air support had failed and that it had “temporarily” lost contact with the city’s officials.

A recently evacuated resident said friends and family who had remained in the city told her the Russian flag was already flying in some districts, and that there were tanks and non-Ukrainian soldiers on the streets.

A city official, Pyotr Andryuschenko, said on Saturday that Russian forces had taken “between 4,000 and 4,500 Mariupol residents forcibly across the border to Taganrog,” a city in southwestern Russia. Given the chaos in the city, his claim could not be independently verified, though it was supported by testimony from others who had recently fled.

If Russian forces should seize Mariupol, it would be one of the few major cities they have taken and would give them control of much of the southern coast of Ukraine. Sounding a note of resignation, an adviser to the Ukrainian president said on Friday, “There is no chance for lifting the siege of Mariupol.”

The Russian advances in Mariupol have hindered frantic efforts to find survivors in the rubble of a theater that was nearly leveled in an attack on Wednesday. About 130 people have been rescued from the theater, according to Ukrainian officials, who estimated that hundreds of people, perhaps as many 1,300, could still be trapped in the basement.

Russian forces struck the theater, even though the word “children” was written in large white letters on the ground at either end of the building.

“These are barbaric methods,” said Mr. Zarubin, the Mariupol doctor. “I think this is genocide.”

Even as Russian forces pushed into Mariupol, the Ukrainian army claimed to have taken back towns and villages around Kherson, one of the first cities to fall. To the west, the Ukrainian military’s defense of the strategic city of Mykolaiv continued to hold, preventing a Russian advance on Odessa, a major port on the Black Sea. And a bloody battle for Kyiv, the capital, loomed, as Ukrainian and Russian troops engaged in fierce fighting in the suburbs.

In western Ukraine, a region that had been largely spared the heavy fighting in the south and east, Russian forces escalated their attacks on military targets.

Russia’s claim that it had used a hypersonic missile in Ivano-Frankivsk came one day after its rockets hit a warplane repair plant near Lviv, rattling a western city that has been a haven for Ukrainians fleeing more embattled areas. And last Sunday, a Russian airstrike hit a military base just 11 miles from the border with NATO member Poland.

The fighting across Ukraine has led to to the fastest-moving exodus of European refugees since World War II. More than a fifth of the 44 million people who were living in Ukraine before Russia invaded last month have been internally displaced or have fled to other countries, according to estimates from the United Nations.

And for those who remain in the country, millions face a daily struggle for survival as cities hard hit by fighting run low on food and clean water, and lack medical care, heat and electricity.

On Friday, the United Nations completed its first convoy of humanitarian aid to the hard-hit city of Sumy in eastern Ukraine, delivering what it said were medical supplies, food and water for 35,000 people.

“We hope this is the first of many shipments delivered to the people trapped by fighting,” said Amin Awad, the crisis coordinator for the United Nations in Ukraine.

As NATO braces for any possible incursion into allied countries, the Norwegian authorities reported that four U.S. Marines had been killed on Friday when their Osprey aircraft crashed there during NATO exercises.

The cold-weather exercises, involving 30,000 troops from 25 countries in Europe and North America, were announced more than eight months ago before the start of the war in Ukraine, NATO said. But they have taken on greater significance in the aftermath of the invasion.

With no clear diplomatic solution to the war, and the death toll rising by the day, there has hardly been time to mourn the losses in Ukraine. But in the heart of Lviv, residents created a striking memorial: 109 empty strollers parked on the cobblestones in Rynok Square, meant to symbolize the 109 children who Ukrainian officials say have been killed by Russian bombardments.

A photo of the memorial, posted by Lviv’s mayor, Andriy Sadovy, was shared widely on Facebook.

“This is the terrible price of war that Ukraine is paying today,” Mr. Sadovy wrote.

Michael Levenson reported from New York, Marc Santora from Warsaw and Valerie Hopkins from Lviv, Ukraine. Reporting was contributed by Michael Schwirtz from Odessa, Ukraine; Victoria Kim from Seoul and Anatoly Kurmanaev from Mexico City.