KYIV, UKRAINE -- Russian forces fired cruise missiles at the southern Ukrainian city of Odesa and shelled the eastern Donetsk region early Wednesday, killing at least six people and damaging dozens of homes, regional Ukrainian officials said.

Russian forces have stepped up aerial strikes in their nearly 16-month war, a Ukrainian military spokesman said. The country's armed forces, meanwhile, have reported limited gains in the early stages of a counteroffensive to take back the nearly one-fifth of Ukraine's territory that is under Russian control.

The grinding Ukrainian advance is pressing slowly ahead, Ukraine's Deputy Defence Minister Hanna Maliar said Wednesday. Western analysts and military officials say the effort to dislodge entrenched, powerfully armed and large numbers of Russian troops could take years.

Ukrainian troops have advanced 200-500 metres (650-1,600 feet) at various sections of the front line around the Donetsk city of Bakhmut and 300-350 metres (980-1,150 feet) in the southern Zaporizhzhia region, Maliar claimed. Ukrainian forces have managed to make gains despite the Russian edge in artillery and air power, she said.

It was not possible to verify the battlefield claims.

In Odesa, three food warehouse employees were killed in a strike that also damaged homes, shops and cafes in the city's downtown, the regional administration said on Facebook. Another 13 people were injured.

Search teams were looking for possible survivors under the rubble of the warehouse, it said.

The attack on the port city, launched from the Black Sea, was the second in a week and involved four Kalibr cruise missiles, three of which were intercepted by air defenses, the administration said.

In eastern Ukraine, Donetsk province governor Pavlo Kyrylenko wrote on Telegram that at least three people died after shelling destroyed seven homes and damaged dozens more in the cities of Kramatorsk and Kostiantynivka.

Ten towns and villages along the front line in Donetsk were struck as Kyiv's troops slowly advance, according to Ukraine's presidential office.

A missile hit the Ukrainian-controlled city of Kramatorsk, where Kyiv's forces are headquartered, killing two civilians and wounding two others while damaging 29 homes, the presidential office said. Russian shelling of Kostiantynivka killed one civilian, with 57 houses damaged, it added.

Andriy Kovalov, a spokesperson for the General Staff of Ukraine's armed forces, said the Russian military increased missile and aerial strikes as Kyiv's forces intensify attacks along the war's 1,000-kilometre (600-mile) front line and claim some modest gains at the beginning of their counteroffensive.

In a briefing, he said strikes on the Kharkiv, Donetsk and Kirovohrad regions, in addition to the Odesa region, involved Kh-22 cruise missiles, sea-launched Kalibr cruise missiles, and Iranian-made Shahed drones. Nine were intercepted.

Kovalov said Ukrainian forces had made advances in several sections and fighting was continuing in or near at least two Donetsk province communities.

Britain's Ministry of Defence, which has regularly issued updates on the conflict, wrote on Twitter that southern Ukraine "has often been more permissible for Russian air operations" compared with other parts of the front.

Separately, the mayor of the central city of Kryvyi Rih, President Volodymyr Zelenskyy's hometown, said the death toll from a Russian strike that hit an apartment building a day earlier had risen to 12.

Ukrainian authorities continued to rescue people from the flooded areas of southern Ukraine's partially Russian-occupied Kherson region following the destruction of the Kakhovka dam last week.

A total of 28 settlements on the Ukrainian-held western bank of the Dnieper River remain under water, and nearly 2,800 people have been taken to safety so far, the presidential office said, adding that the rescue effort was taking place under relentless Russian shelling.

The Ukrainian-controlled areas of the Kherson region came under artillery fire 57 times over the past 24 hours, the presidential office said.

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Associated Press writer Yuras Karmanau in Tallinn, Estonia, contributed to this report