(Adds Ukraine president, United Nations spokesman comments)
* Sievierodonetsk mayor says some evacuations still possible
* Ukrainians holding out against Russian assaults
* Ukraine pleads for more heavy weapons from West
By Natalia Zinets
KYIV, June 14 (Reuters) - Ukraine said its forces were still trying to evacuate civilians from embattled Sievierodonetsk on Tuesday after Russia destroyed the last bridge to the city, Moscow's main target in its battle for control of eastern Ukraine.
Ukrainian forces were holding out against the Russians after ground changed hands several times over the past few weeks, enduring what President Volodymyr Zelenskiy described as "painful" losses.
"But we have to hold strong ... The more losses the enemy suffers, (the) less strength it will have to pursue its aggression," Zelenskiy said in an address on Tuesday night.
Although Sievierodonetsk's last bridge over the Siverskyi Donets river had been felled, evacuations were still being carried out "every minute when there is a lull and there is a possibility of transportation," Mayor Oleksandr Stryuk said.
"Every possible chance is taken," Stryuk said.
Ukraine says more than 500 civilians are trapped inside Azot, a chemical factory where its forces have resisted weeks of Russian bombardment and assaults that have reduced much of the city to ruins.
Luhansk regional governor Serhiy Gaidai said shelling on Azot was so powerful that "people can no longer stand it in the shelters, their psychological state is on the edge."
Russia said it would give Ukrainian fighters holed up in the chemical plant a chance to surrender on Wednesday morning. Fighters should "stop their senseless resistance and lay down their arms" from 8 a.m. Moscow time (0500 GMT), said Mikhail Mizintsev, head of Russia's National Defence Management Centre.
Civilians would be let out through a "humanitarian corridor", he said.
Both sides claim to have inflicted huge casualties in the fighting over the city, Russia's principal battleground focus.
Ukraine still holds Lysychansk, Sievierodonetsk's twin city on higher ground on the western bank of the river. But with all the bridges now cut, its forces acknowledge a threat that they could be encircled in Sievierodonetsk.
Damien Megrou, spokesperson for a unit of foreign volunteers helping to defend Sievierodonetsk, said there was a risk of leaving "a large pocket of Ukrainian defenders cut off from the rest of the Ukrainian troops" - as in Mariupol, the Black Sea port that surrendered last month after months of Russian siege.
The battle for Sievierodonetsk - a city of barely more than 100,000 people before the war - is now the biggest fight in Ukraine as the conflict has shifted into a war of attrition.
Kyiv has said 100-200 of its soldiers are killed each day, with hundreds more wounded.
Russia gives no regular figures of its own losses but Western countries say they have been massive, as Moscow has committed the bulk of its firepower to delivering one of President Vladimir Putin's stated objectives: forcing Kyiv to cede the full territory of two provinces, Luhansk and Donetsk, collectively known as the Donbas.
Momentum in Sievierodonetsk has shifted several times over the past few weeks - with Russia concentrating its overwhelming artillery firepower on urban districts to obliterate resistance, then sending in ground troops vulnerable to counter-attacks.
Bigger battles could lie ahead for the wider Ukrainian-held pocket of the Donbas, nearly all on the opposite bank of the river which Russian forces have found difficult to cross. Ukraine says Russia plans to assault Sloviansk from the north and along a front near Bakhmut to the south.
In Donetsk province, critical infrastructure including homes, schools, hospitals and markets have been attacked over the past week, United Nations spokesman Stephane Dujarric told reporters in New York.
"This has made life nearly unbearable for people who are also facing severe water shortages, and at times are unable to leave their homes for days on end due to the fighting," Dujarric said.
Ukrainian officials renewed their pleas for the West to send more and better artillery as well as tanks, drones and other heavy weapons.
Deputy Defence Minister Hanna Malyar told a briefing that Ukraine had received only 10% of the arms it needed from its allies.
Western countries have promised NATO-standard weapons - including advanced U.S. rockets. But deploying them is taking time.
"What we don't have enough of are the weapons that really hit the range that we need to reduce the advantage of the Russian Federation's equipment," Zelenskiy told an online press briefing.
Beyond the Donbas, Ukrainian officials hope that Russia's focus on capturing the east will drain its forces from other areas and pave the way for counter-attacks to recapture territory.
Ukraine retook the area around its second largest city Kharkiv in May and has reported small but steady gains in recent days in the south, the biggest swathe Russia retains of the territory it seized after its invasion in February.
Serhiy Khlan, adviser to the head of the southern, mainly Russian-occupied Kherson province, said Ukraine was having tactical success retaking territory there for a second straight week.
Troops had already advanced 5 km (three miles) from Tavriysk, a town on the south bank of the Dnipro river east of Kherson city, and were gradually advancing, Khlan said.
Reuters was unable to verify any reports from the area.
(Reporting by Reuters bureaux; Writing by Peter Graff and Angus MacSwan and Rami Ayyub; Editing by Gareth Jones, Mark Heinrich and Grant McCool)