President Volodymyr Zelensky of Ukraine visited the embattled eastern town of Avdiivka, his office said on Tuesday, after the Kremlin announced that President Vladimir V. Putin of Russia had again traveled to occupied areas of Ukraine near the front line, as both leaders sought to display strength and rally their troops.

The split-screen images came ahead of an anticipated Ukrainian counteroffensive to take back territory seized by Russian troops. The trips also occurred as the battle for the key eastern city of Bakhmut intensified, with Moscow launching airstrikes and attacking from several directions simultaneously, a Ukrainian general said.

Mr. Putin visited the southern region of Kherson and the eastern Luhansk region, the Kremlin’s spokesman, Dmitri S. Peskov, said. It was the Russian leader’s second trip close to the front line in a month. Military analysts say that Ukraine might target Kherson and Luhansk in its counteroffensive.

Mr. Zelensky announced his visit on Tuesday in a post on the social messaging app Telegram. But it was not immediately clear when Mr. Putin made his trip, though Mr. Peskov and the Kremlin said it was on Monday. The Russian leader appeared in different clothes in two parts of his trip and could be heard in video footage released by Moscow speaking with his commanders about the upcoming Easter, which the Orthodox Church celebrated on Sunday.

His remark, which appeared in a video broadcast by Russian state media, was removed from later versions. Mr. Peskov said that Mr. Putin had meant the Easter season, which lasts several weeks, and insisted that the trip had taken place on Monday.

Mykhailo Podolyak, an adviser to the Ukrainian president, denounced Mr. Putin’s trip, saying on Twitter that it was little more than a “special tour” of an area that Russia’s leader had ruined when he ordered the full-scale invasion of its neighbor in February 2022.

During his trip to the town of Avdiivka — which Russian forces have being trying to seize for more than a year, leaving it in ruins and forcing almost all of its residents to flee — Mr. Zelensky sat with troops and wished them a happy Orthodox Easter, his office said.

The Ukrainian leader made two trips in two straight days to the front line in March, in a show of resolve and to thank soldiers on the front line. He made an unannounced visit near the Bakhmut area, and that same month, he visited areas in the Kherson region ravaged by Russia’s campaign to destroy energy infrastructure.

In December, Mr. Zelensky also went to Bakhmut, which has become a potent symbol of Ukrainian resistance in the face of Moscow’s ceaseless assault.

The two visits this week were announced as fighting has intensified, including in Bakhmut, where the Russian Defense Ministry said on Monday that its troops had captured two more areas in the south and northwest of the city, although the claim could not be verified.

“The Bakhmut sector remains the epicenter of the fighting,” the commander of Kyiv’s ground forces, Gen. Oleksandr Syrsky, said on Tuesday on Telegram.

“Currently, the enemy is increasing the activity of heavy artillery and the number of airstrikes, turning the city into ruins,” General Syrsky said. But Ukrainian forces remain in control, he added.

Fighting also raged in Avdiivka, where Russian attacks have destroyed entire neighborhoods and nearly cut off access to humanitarian aid for its remaining residents. The city once had a population of 30,000, but many have fled. Ukrainian officials estimate about 1,800 residents are refusing to evacuate.

After Mr. Putin’s trip to Kherson, Russian forces a shelled a market in the center of Kherson City, killing one person and wounding at least six others, the head of Mr. Zelensky’s office, Andriy Yermak, wrote on Telegram.

In all, Russian forces had launched 342 shells at the Kherson region in the previous 24 hours, Oleksandr Prokudin, the regional military administration head, wrote on Telegram on Tuesday.

Mr. Putin last visited occupied areas of Ukraine a month ago, traveling to Crimea and the city of Mariupol — a day after an international court issued a warrant for his arrest, accusing him of war crimes.

This time, he appeared to have traveled under a veil of secrecy and without his usual number of staff, which typically includes photographers and video operators. The Kremlin released grainy, shaky video footage of the trip.

Mr. Putin took a military helicopter to Kherson, though it was unclear where exactly he visited. Photographs and video released by Russian state media showed him emerging from a helicopter that had landed in a rural area on the eastern side of the Dnipro River.

Mr. Putin also visited Russian military headquarters in Luhansk, the Kremlin said in a statement. The Russian president was not accompanied by his defense minister, Sergei K. Shoigu, or by the chief of the general staff of the military, Valery V. Gerasimov, according to footage broadcast by the state media.

In opening remarks, Mr. Putin told military commanders, “It was important for me to listen to your opinion about the current situation, to share information.”

Russia seized Kherson City, the regional capital, in March last year, when its troops advanced north from Crimea and crossed over the Dnipro with almost no opposition. It was the only time since Moscow’s full-scale invasion that it had seized a regional capital.

That success lasted but a few months.

Last summer, the Ukrainian government selected the Kherson region for its first major counteroffensive. Armed with military aid from the United States and other allies, it targeted Russian forces and military infrastructure in the province with rockets and fought intense battles across the province on both sides of the river.

Moscow had stationed tens of thousands of troops in Kherson City, but with key bridges destroyed or impassable, they became exposed. Before a full-scale battle for the city began, Russian commanders ordered a withdrawal to the east river bank in November. Ukrainian forces then entered the city of Kherson and retook much of the Kharkiv region in September.

Having retreated, Russian forces are still pounding Kherson City and surrounding areas held by Ukraine with a daily barrage of rocket fire, killing civilians, damaging towns and villages and making the resumption of normal life virtually impossible.

Ukrainian officials and military experts say Russia has been building up its forces in the Kherson area, laying mines, increasing troop numbers and constructing defensive barriers in anticipation of Ukrainian attacks.

Kyiv has kept the location and timing of any counteroffensive under wraps, but a campaign to retake land in the south could, if successful, mean that Crimea, which Moscow illegally annexed from Ukraine in 2014, becomes divided from the territory that Russia holds in eastern Ukraine.

Ukrainian troops have also been squeezed into defending a section of the western part of Bakhmut as Russian forces have made steady advances on the city in months of bloody fighting.

“This is where the enemy is concentrating most of its efforts and is determined to take control of the city at any cost,” General Syrsky said.

Russia began its assault on Bakhmut last summer, and the battle has led to heavy casualties on both sides, though military experts say Russian losses are significantly higher.

If Russian forces capture all of Bakhmut, it would mark their first seizure of a key city in months.

Britain’s defense secretary, Ben Wallace, on Tuesday expressed confidence in Ukraine’s planned counteroffensive, but signaled that the war would most likely continue into next year.

“I’m optimistic that between this year and next year, I think Ukraine will continue to have the momentum with it and a position of strength,” Mr. Wallace told reporters in Washington. “I also think we should be realistic: There is not going to be a single magic-wand moment when Russia collapses.”

Eric Schmitt, Yousur Al-Hlou and Masha Froliak contributed reporting.