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

The United States is standing by the NATO communique released Tuesday and its language around Ukraine joining the alliance, despite criticism from Ukrainian President Volodymyr Zelenskyy over the lack of a timeline to do so.

"The United States clearly joined with NATO allies in agreeing to a strong positive message reaffirming that Ukraine will become a member of the alliance," U.S. National Security Council Senior Director for Europe Amanda Sloat said during a press briefing in Lithuania's capital on Wednesday morning, on the final day of a high-stakes NATO summit. "And as the communique has made clear, as the president has spoken to directly in the past, we recognize that Ukraine has already made significant progress in terms of reforms. That was part of what led to allies making the decision to say that the Membership Action Plan was no longer required for Ukraine."

"But as both the president has said and as the communique made clear, there is still the need for Ukraine to take further democratic and security sector reforms," she added.

Sloat told reporters that the U.S. has been and would continue to work with Ukraine both bilaterally and through the NATO alliance to ensure that the reforms required to join the alliance are met.

When asked to respond to Zelenskyy's criticism that the lack of a timeline was "unprecedented and absurd," Sloat defended the agreement as a significant one.

"I would agree that the communique is unprecedented, but I see that in a positive way. We joined with allies yesterday in agreeing to a very strong, positive message. We reaffirmed that Ukraine will become a member of the NATO alliance," she said, arguing that removing the Membership Action Plan requirement for Ukraine was a "very significant" step on NATO's part.

-ABC News' Molly Nagle