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 prepared to defend every inch of NATO territory," National Security Advisor Jake Sullivan said Wednesday, amid Russia's threats of military retaliation if Sweden and Ukraine officially join the alliance.

"At the end of the day, Russia does not get a vote in who joins NATO," Sullivan told ABC News' George Stephanopoulos during an interview Wednesday on "Good Morning America."

"If Russia does choose to test Article 5, President Biden has said he is prepared to defend every inch of NATO territory," he added. "And later tonight, when he addresses a large crowd in Vilnius in a speech about his vision for Euro-Atlantic security and for American leadership, he will reinforce this point about the sanctity of Article 5."

Article 5 of the NATO treaty states that the signed parties "agree that an armed attack against one or more of them in Europe or North America shall be considered an attack against them all."

Sweden and Finland, which have historically embraced neutrality, applied for NATO membership together last year despite warnings from Moscow against doing so, as Russian forces invaded Ukraine. Finland officially joined the alliance in April following Turkish ratification, but Turkey and Hungary have yet to ratify Sweden’s membership. NATO Secretary-General Jens Stoltenberg announced Monday that Turkish President Recep Tayyip Erdogan has agreed to send Sweden’s NATO accession documents to Turkey’s Parliament for approval "as soon as possible," one of the last steps in the membership process.

Meanwhile, Ukraine is pushing hard for NATO membership as Russia’s war presses on. Stoltenberg announced Tuesday that the bloc "will issue an invitation for Ukraine to join NATO when allies agree and conditions are met," pledging to speed up the process by eliminating bureaucratic hurdles, but without offering a timeline.

"All 31 allies got together yesterday to say with one voice that Ukraine's future is in NATO. The real question is just how we work down the pathway to get them in," Sullivan told ABC News on Wednesday. "So the signal that NATO sent yesterday was very positive to Ukraine about its future prospects for joining NATO, and we will work in terms of the democratic and security sector reforms that are necessary. And in the meantime, we're not just going to sit around. We're going to provide Ukraine with the weapons and military assistance it needs to defend its territory against Russia and to deter future aggression from Russia."

Ukraine has requested F-16 fighter jets and long-range missiles to aid its counteroffensive against Russian forces. Sullivan said the U.S. has already taken steps with NATO allies to begin the training of Ukrainian pilots on F-16s, which "will take some time," and then the aircraft will be transferred to Ukraine "likely from European countries that have excess F-16 supplies." As for the long-range missiles, Sullivan said the U.S. will continue to look into the issue and discuss it with Ukraine.