So far, Ukraine’s push to retake territory captured by Russia appears to be focused on the eastern and southern Donetsk region, near the boundary with the Zaporizhzhia region. Kyiv’s forces are apparently trying to drive a wedge into Russian-held territory between the Russian border and occupied Crimea.

There have been reports of fierce fighting near the Mokri Yaly River and the string of small settlements that run along it. Ukraine has also accused Russian forces of destroying a small dam there, with the aim of slowing the counteroffensive, days after the destruction of the Kakhovka dam drained a huge reservoir and caused widespread flooding.

How much territory does Ukraine say it has reclaimed?

Over the weekend, Ukraine said that it had retaken several farming villages along the Mokri Yaly River, including Storozhove and Blahodatne. The recaptured territory covers about 35 square miles, a senior Ukrainian official claimed on Monday.

But the significance of those gains remained to be seen, and by Wednesday, Ukraine did not appear to have broken through any of Russia’s defensive lines. It could take weeks or months to gauge the success of the military actions, analysts say.

In its daily update on Wednesday, the Ukrainian military’s general staff said fighting was underway in several villages in western Donetsk and eastern Zaporizhzhia. Hanna Malyar, a deputy defense minister, said Ukrainian troops had “partial success” in fierce fighting, but she did not claim any significant gains.

How is Russia responding?

President Vladimir V. Putin, speaking to Russian war correspondents and military bloggers, acknowledged on Tuesday that his forces had suffered some losses in June. But he denied Ukraine’s claims that its forces had made progress on the battlefield and claimed that Ukrainian forces had suffered significantly more losses of military equipment than the Russians.

Yevgeny V. Prigozhin, the head of the Wagner paramilitary group that has fought for Russia in Ukraine, offered a different view, saying on Tuesday that Ukrainian forces had most likely reclaimed more than 100 square kilometers (about 38 square miles). Mr. Prigozhin has often been at odds with Russian military officials.

On Tuesday, Russian officials and military bloggers said that Russian attack helicopters had struck a Ukrainian position near the town of Velyka Novosilka, in the area of the villages Ukraine claims to have retaken.

How great are Ukraine’s losses?

Although Ukraine is tight-lipped about its casualties and equipment losses, it is clear that its forces are facing heavy defenses that Russia has built up over months, including minefields, trenches, anti-tank ditches, a well-equipped air force, and artillery fire. And as soldiers venture forward, military analysts say, they move out of the range of their own army’s air defenses and electronic jamming systems, leaving them vulnerable to Russian air attacks.

Several American-made Bradley fighting vehicles were abandoned by Ukrainian troops or destroyed in the early days of the counteroffensive, based on videos and photographs posted by pro-war Russian bloggers and verified by The New York Times. Such losses, as well as casualties, are to be expected in the initial phase of the counteroffensive, military analysts say.

So far Ukraine has not committed the bulk of its forces, including Western-trained units, to any one place, instead trying to probe Russian lines for weaknesses. At earlier points in the war, the true toll of intensified fighting often became clear only weeks later, with an uptick in military funerals in communities across Ukraine.