Kyiv awaited supplies of air defence munitions amid a shortage that has left it exposed to Russian attacks.
Two glide bombs hit a crowded area in the northern city of Sumy overnight, a frequent Russian target, killing five people and injuring 30, authorities said.
One bomb hit a bus stop and pictures showed a yellow bus with one of its sides ripped off. Other pictures posted online showed damage to apartment building facades.
In a border district of Sumy region, where the Kremlin wants to expand a buffer zone, a local official said a man was killed after stepping on an explosive device
Glide bombs also injured 10 in the southeastern city of Zaporizhzhia, the regional governor said.
Two people were killed and another wounded earlier in the day by a missile strike on the southern port city of Odesa.
Zelenskiy, speaking in his nightly video address, said Ukraine needed its allies to ensure weapons supplies were provided more promptly.
"I am preparing changes in Ukraine's diplomatic efforts. We need a new level of co-operation with our partners to ensure that agreements on arms supplies are fulfilled," Zelenskiy said.
"Agreements reached by national leaders must be implemented much more quickly and completely."
This, he said, applied to an agreement reached with US President Donald Trump this week to grant Ukraine a licence to produce its own Patriot interceptor missiles.
Ukraine is critically low on munitions for its Patriot air defence systems and has been largely unable to down ballistic missiles, which travel at several times the speed of sound, over the past month.
It has pleaded with allies for greater supplies of those munitions and has also pushed Europe to work with Kyiv on its own anti-ballistic air defence system.
Earlier on Saturday, a drone hit a civilian enterprise in the eastern city of Kharkiv, wounding seven.
The head of Russian-controlled areas of Donetsk region in Ukraine's east, Denis Pushilin, said one person was killed when a drone hit his car and nine were injured when a bus came under a drone attack near the Sea of Azov.
Twelve people, meanwhile, had been wounded in the capital Kyiv during an overnight attack that used ballistic and cruise missiles as well as drones.
"Civilian infrastructure was hit even before the air raid alert was issued," Zelenskiy said earlier of the attacks on Kyiv.
"Our defenders managed to shoot down most of the targets, but not the ballistic ones."
Russia launched six ballistic missiles, another six cruise missiles and 121 drones in the overnight attacks including against Kyiv, Ukraine's air force said on Saturday morning, before the strikes on Odesa, Kharkiv and Sumy took place.
It said it downed at least two cruise missiles and 111 drones.
Russia has stepped up attacks on the capital in recent weeks. So far this month, strikes on Kyiv and the surrounding region have killed more than 60 people.
Kyiv, in turn, has been pressuring Russia's military logistics in occupied southern Ukraine, seeking to deprive Russian forces of fuel and munitions by conducting strikes on trucks and vessels deep behind the front lines.
Kiev Mayor Vitali Klitschko said the recent heavy Russian air attacks against the capital were unprecedented and are a result of Ukraine's newfound battlefront strength.
"The attacks with Russian ballistic missiles are worse than ever before," he told the tabloid Bild's Sunday edition. These amount of a further escalation of Moscow's war against Ukraine, he said.
Within a week, the Russian military has launched three heavy aerial attacks on Kiev. In addition to drones, it used cruise missiles and ballistic missiles.
The consequences for the Ukrainian capital have been devastating. Dozens of people were killed and more than 100 were injured. The impacts damaged or completely destroyed many residential buildings in Kiev.
Ukraine's drone forces chief Robert Brovdi said his units had struck 21 fuel tanker vessels in the Sea of Azov overnight, as well as seven other cargo and support ships, bringing the total number of vessels struck this week to 76.
Zelenskiy has said the aim of the drone campaign is to bring Russia to the negotiating table, although Russia's Vladimir Putin has not yet shown any willingness to soften his position.
One person was killed in a drone attack on four vessels, including a tanker carrying methanol, in Taganrog Bay on the Sea of Azov, Russian authorities said on Saturday.
with dpa
