The general leading Ukraine's counteroffensive along the southern front line says his forces have broken through in Verbove -- and predicts an even bigger breakthrough to come.

"On the left flank [near Verbove] we have a breakthrough and we continue to advance further," Oleksandr Tarnavsky told CNN Senior International Correspondent Frederik Pleitgen during an interview on Friday, though he conceded his troops were moving slower than anticipated.

"Not as fast as it was expected, not like in the movies about the Second World War," he said. "The main thing is not to lose this initiative (that we have). And, well, not to lose it in practice, with actions."

The general's claim is the latest indication by Ukrainian officials that inroads are being made on the southern front in the war with Russia.

Ukrainian forces claimed in recent weeks to have penetrated the "first line" of Russian strongholds in the Zaporizhzia region, in a sign that Kyiv was edging closer to Moscow's sprawling network of fortified trenches along the southern front.

Russian-appointed officials in occupied Zaporizhzhia have given a different picture of the fighting. CNN is unable to verify the battlefield reports of either side. However, open-source analysis of available video suggests that some Ukrainian units have crossed through an important line of Russian defenses near the village of Verbove.

Ukraine's long-term goal is to break Russia's "land bridge," which links territory it holds in the east with annexed Crimea.

At the beginning of September Ukrainian forces said they had taken the village of Robotyne and were pushing east toward the village of Novoprokopivka. Soldiers said they expected battles for control of high ground to the south and east of the village as they approached the next layer of Russian defenses. Verbove is a few miles east from Robotyne.

However, Tarnavsky told CNN he believed the big breakthrough of the counteroffensive would be if Ukraine could take Tokmak, a strategic hub for Russia, which is its first major target in its fight in the south.

CNN reported earlier this week that the Ukrainian forces were still some 20 kilometers from Tokmak, struggling to break through the multiple layers of Russian defenses.

"I believe yes [there will be a big breakthrough]," Tarnavsky said. "I think it will happen after Tokmak. At the moment (the Russians) are relying on the depth of their defensive line there."

Rather than the 'Surovikin line' -- a defensive line built on the orders of former General Sergey Surovikin, while he was in charge of Russian forces in Ukraine -- Tarnavsky says the bigger issues are the "crossroads, tree lines and minefields between the tree lines."

"[There's] a combination of small harmful enemy defense groups that currently are planted very precisely and competently," he said. "But the actions of our fighters force them to slowly pull back when they face our assault squads."

The general conceded that for the counteroffensive to be a success, Ukrainian forces need to at least reach the city of Tokmak.

"Tokmak is the minimum goal," he said. "The overall objective is to get to our state borders."

'Not as fast as expected'

Despite conceding the advance wasn't going as "fast as expected," Tarnavsky said winter would not slow down the counteroffensive.

Intense rains in the fall can make the ground in Ukraine soggy and make movement with heavy machinery, like tanks, more difficult, but Tarnavsky says Ukraine's forces move in small groups, mostly on foot.

"The weather can be a serious obstacle during advance, but considering how we move forward, mostly without vehicles, I don't think [the weather] will heavily influence the counteroffensive," Tarnavsky told Pleitgen in the interview.

"Right now, neither the enemy nor us uses large formations, companies, battalions or brigades. We use assault squads, groups of 10 to 15 men," he explained. "They conduct titanic work of concentrating enemy fire on them and using all the means they have to use to survive."

One of the reasons for the slow advance, the general said, was the fact that Russia had been able to learn some lessons from other Ukrainian offensives.

"The Russians are learning quite fast, as they don't have any other choice. If they don't learn, they will be defeated sooner," he explained. "I wouldn't say they are adapting to our actions, as we also change our tactics."

He said Ukraine had also been adapting to using Western equipment, as well as Western tactics.

When asked about rising resistance in the West to continuing weapons supplies to Ukraine, especially in the US, where some have voiced doubt about Kyiv's chances of success, Tarnavsky said he respected their view.

"Let it be the opinion of the skeptics. It's not a competition and we don't have a choice," he said. "I understand somebody may have imagined large scale counteroffensive operations. But times are changing, enemy tactics are changing, weapons change."

"We have one goal -- liberation of our territories. However hard it is we will keep on working. And I want to thank even the skeptics, their criticism also influences our task's success," he said.

He thanked Ukraine's Western allies for their continuous support and pledged to treat the tanks and armored vehicles they have supplied with great care.

Wagner, 'here and there'

During the interview, the general also commented on the current state of the mercenary group Wagner.

He said Wagner fighters still pop up "here and there" on the front line following the death of the group's former boss Yevgeny Prigozhin, who died in a plane crash last month after leading a failed insurrection that posed the biggest threat to the rule of Russian President Vladimir Putin in decades.

A deal struck to de-escalate the crisis was supposed to have sent Prigozhin and his fighters to neighboring Belarus, but there have been reports of Wagner fighters redeploying to Kherson.

Asked about those reports, Tarnavsky said, "In some directions, I can't say whether it's the Kherson region, or our direction, or somewhere else, they do pop up. The fact is that their badges appear here and there -- that's been constant."

Tarnavsky went on to say his men usually speculated about the presence of Wagner whenever their enemy started behaving more competently.

"We suspect the presence of Wagner if we see the deterrence of our offensive forces with the involvement of a category of military personnel who perform these tasks in a more interesting (more non-standard) way," he explained. "This makes one think: "Maybe it's Wagner has showed up?""

"But there is no such unit in my area of the front today," he added. "Regarding Kherson direction -- I can't say."

Ukraine recently launched a missile attack on the headquarters of Russia's Black Sea Fleet in Sevastopol, in the Crimean peninsula, which was illegally annexed by Moscow's forces in 2014.

Tarnavsky told CNN that strikes on Crimea are important for the success of Kyiv's counteroffensive.

"The success of offensive operations is not only about destroying the enemy in front of you, it's also about destroying places of concentration of equipment, personnel and especially destroying the command centers," he said, adding that it can create a "a mess on the battlefield."

The general added striking Crimea was especially important as a rallying cry for Ukrainian forces and to boost morale.

"It helps us but it also gives us hope for the future," he said.