At least eight people are dead and 13 others are injured after a Russian rocket struck the Ukrainian town of Lyman in the Donetsk Region on Saturday morning, according to the head of the regional military administration, Pavlo Kyrylenko.

The attack struck a neighborhood with private residences, damaging a home and a store.

As of Kyrylenko's update, police and paramedics were still on site to provide necessary medical assistance.

-ABC News' Tatyana Rymarenko

Russian shelling on the town of Pervomayskyi, located in the region of Kharkiv, has injured 43 people, including 12 children, Oleh Synyehubov, the head of the Kharkiv Regional State Administration, said .

A residential building was damaged and multiple cars were destroyed in the shelling, according to Synyehubov.

-ABC News' Fidel Pavlenko

Ukrainian Armed Forces have yet to reach their "full potential," but two top generals exclusively told ABC News the counteroffensive is "going to plan" despite concerns from Western analysts that Ukraine is not making enough progress.

Gen. Oleksandr Tarnavskyi, who is leading the counteroffensive in the south, told ABC News' chief global affairs correspondent Martha Raddatz that the situation is "stable."

"The main thing is that we haven’t reached our full potential," Tarnavskiy said.

Ukrainian troops launched the counteroffensive a month ago, attacking on multiple axes on the southern frontline in Zaporizhzia using Western-supplied vehicles. Ukraine succeeded in piercing Russian lines at two points, liberating a string of villages, but has since been locked in ferocious fighting

Gen. Oleksandr Syrskyi, commander of the Ukrainian ground forces, told Raddatz that last month's failed rebellion by the Putin-backed PMC, the Wagner Group, wouldn't impact the fighting on the ground.

"It would be better for us if there were some negative consequences in Russia itself, but it doesn't matter for me," he said.

Asked if he was confident of retaking the key city of Bakhmut, Syrskyi said, "Yes, of course. I’m sure."

Both sides have suffered heavy casualties, according to Western officials, particularly in the south where Russia has so far conducted "relatively effective defensive operations" in the southeastern Zaporizhzhia province, the U.K. Ministry of Defense reported.

"The enemy is suffering eight times or even 10 times higher losses, especially with the number of killed troops," Syrksyi said.

-ABC News' Guy Davies and Meghan Mistry

At least two people were killed and 19 others injured from a Russian drone attack in northern Ukraine, the Sumy Regional Military Administration said.

An administrative building and two apartment buildings were damaged in the four drone strikes, officials said.

Four people have been hospitalized, according to officials. Two of the patients were listed in intensive care.

Ukrainian President Volodymyr Zelenskyy said in his daily evening address that the lack of air defense systems is part of the reason buildings were hit.

"The enemy is taking advantage of this," he said.

-ABC News' Ellie Kaufman

Ukrainian troops have reportedly seized a small village on the Russian-occupied eastern bank of Dneipr after crossing the river from the liberated city of Kherson, according to Russian accounts on Tuesday.

A small Ukrainian force has managed to dig in to the village of Dachi after making a landing there a few days ago and are trying to expand the beachhead. Ukrainian troops in small boats crossed the river and landed at the base of the ruined Antonivskiy Bridge, which they destroyed last year.

Russian military bloggers reported very heavy fighting on Tuesday, saying Russian airborne units have been trying unsuccessfully to dislodge the Ukrainians for the past four days. Russian aviation and heavy artillery have been firing on the Ukrainian position.

So far, the Russian side claims some 70 Ukrainian soldiers are dug in, covered by intense artillery fire from across the river. The Russian accounts say Ukraine is trying to move reinforcements across. Video released by Ukraine also shows a Russian APC being destroyed in the village of Oleshkjy, further south, indicating the bank south of Kherson is now contested.

Ukrainian troops have also managed to advance and liberate Rivnopil, a village on the Zaporozhzhia front, breaking through after more than two weeks of fighting. It's notable because the Russians had been fighting hard to hold it.

Taken together the advance there, the landing in Kherson and advances near Bakhmut are small signs the Ukrainian counteroffensive may be starting to pick up steam and the Russians are coming under growing pressure.

-ABC News' Patrick Reevell