"The second convoy of Ukrainian supplies has just left... three from Chornomorsk and one from Odessa," Kyiv's infrastructure ministry wrote on Telegram.

It said the Mustafa Necati, the Star Helena, the Glory and the Riva Wind were carrying "around 170,000 tonnes of agriculture-related merchandise."

Meanwhile, Moscow and Kyiv traded accusations over who bombed the Zaporizhzhia nuclear site in southern Ukraine, Europe's largest atomic power complex which has been under Russian control since the early days of the February 24 invasion.

The recent fighting at the plant prompted UN's nuclear watchdog, the International Atomic Energy Agency (IAEA) to warn of "the very real risk of a nuclear disaster."

On Sunday, Russia's occupying authorities in the town of Enerhodar where the plan is located said the Ukrainian army overnight "carried out a strike with a cluster bomb fired from an Uragan multiple rocket launcher."

The projectiles fell "within 400 metres of a working reactor," Russia's state news agency TASS reported.

The strike damaged some administrative buildings and fell in a "zone storing used nuclear fuel".

However Ukraine's state nuclear energy company Enerhoatom that operates the plant said the "Russian occupiers once again fired rockets at the site of the Zaporizhzhia nuclear power plant and the town of Enerhodar."

"One... employee was hospitalised with shrapnel wounds caused by the explosion," it said in a statement.

AFP was not able to confirm the allegations from an independent source.

On Saturday, Enerhoatom, had already said parts of the facility had been "seriously damaged" by military strikes the previous day and one of its reactors forced to shut down.

The prospect of the giant complex being seriously damaged in the fighting has set off alarm.

"Any military firepower directed at or from the facility would amount to playing with fire, with potentially catastrophic consequences," IAEA Rafael Grossi said on Saturday.

- 'Sign of hope' -

The renewed shipments of Ukrainian grain to help ease global food shortages and bring down prices nevertheless offer a small glimmer of hope as the war enters its sixth month.

Ukraine, one of the world's largest grain exporters, had been forced to halt almost all deliveries in the wake of Russia's invasion, sending global food prices soaring and making imports prohibitively expensive for some of the world's poorest nations.

In Rome on Sunday, Pope Francis welcomed the resumption of grain exports as "a sign of hope" that showed dialogue was possible to end the war.

"I sincerely hope that, following this path, we can put an end to the fighting and arrive at a just and lasting peace."

A bulk carrier had arrived in Chornomorsk on Saturday to be loaded with grain for the first time since Moscow's invasion.

Last Monday, the Sierra Leone-flagged vessel, Razoni, set sail from the Ukrainian port of Odessa carrying 26,000 tonnes of corn in the first departure under the deal that was brokered with the help of Turkey.

Then on Friday, Kyiv said another three ships loaded with grain had also set off for Turkey and markets in Ireland and Britain with a further 13 waiting to depart.