Fierce battles have been reported as Ukraine tries to retake the Russian-occupied southern Kherson region - but military experts have told the BBC it "won't happen quickly".

"Heavy fighting is continuing, our soldiers are working around the clock," said Vitaliy Kim, who heads the neighbouring Mykolaiv region.

Ukraine earlier said it had broken through Russia's first line of defence.

But Russia said Ukrainian troops had been defeated during a failed attack.

The defence ministry in Moscow also said there had been heavy casualties among the Ukrainian forces, but the claims by both Ukraine and Russia have not been independently verified.

Kherson became the first major Ukrainian city to fall into Russian hands in the opening days of Moscow's invasion that began on 24 February.

Top Ukrainian military officials have been tight-lipped about giving too many details about its reported counter-offensive, urging the wider public to be patient.

On Tuesday, Ukraine's Operational Command "South" reported that "positional battles" continued.

It said three key bridges across the Dnipro River had been hit again to make sure they were impassable. In recent weeks, Ukraine has also been using US-supplied Himars precision rocket systems to target makeshift Russian pontoon bridges, as well as command posts and ammunition depots.

This is widely seen as part of a targeted effort to cut off Russian troops on the right (western) bank of the river, making it virtually impossible to send troops and weapons resupplies.

However, officials in Kyiv have been cautioning against any expectations of a quick win, describing the offensive as a slow grinding down of the enemy.

Several military experts interviewed by the BBC seem to agree with this.

"The long-term trend is the gradual weakening of the Russian military capability and the gradual strengthening of the Ukrainian capability with Western help," said Sir Alex Younger, a former head of Britain's MI6 Secret Intelligence Service.

He said the Kherson counter-offensive was "all about the urge of Ukraine to demonstrate they can get on the front foot and in turn underpin the resolve of Ukraine and its backers into what is going to be a difficult winter".

Justin Bronk, a senior research fellow at the Royal United Services Institute (Rusi) defence think tank, told the BBC he believed Ukraine would continue with strikes using Himars as they were proving so successful at cutting off Russian supply lines.

Moving forward, though, Mr Bronk said he expected to see more of an "attritional strategy" by Ukraine, "pushing [Russia] on a broad front, trying to force Russian morale to break and inflict casualties".

Meanwhile, Ukrainian military expert Mykhaylo Zhyrokhov told the BBC: "We shouldn't be expecting a cavalry attack, because the general staff of Ukraine's armed forces doesn't want street fighting [in the Kherson city] - that would result in heavy casualties."

"The Ukrainian strategy is to squeeze the enemy out," he said, warning that this "won't happen quickly".

"It's like a situation with a cornered rat that needs to be offered a way out. Ukraine's military should aim to provide the Russian troops with such a withdrawal route, otherwise [the Russians troops] would be fighting until the last soldier."

"The window of opportunity is until mid-October because after that the mainly field roads in the region will become impassable" because of mud and slush, Mr Zhyrokhov said.

Meanwhile, a spokesperson for the US state department told reporters on Tuesday that senior Russian officials have been tasked with organising an annexation referendum in Kherson.

Sergei Kiriyenko, a top adviser to President Putin, was overseeing efforts to hold a poll "in the coming weeks," Vedant Patel said.

"The sham referenda will attempt to give a veneer of legitimacy to a blatant land grab that would violate the Ukrainian constitution and international law," Mr Patel added.

Elsewhere, in his nightly address Ukraine's President Volodymyr Zelensky has said that intense fighting has been taking place in the eastern Donbas and Kharkiv regions.