Over 7,500 residents have been evacuated as the crisis forces Zelenskyy to cancel his Spain and Portugal visit.
Russia is significantly intensifying strikes on Kharkiv. On Tuesday, Moscow forces hit the Ukrainian border city twice, using guided aerial bombs and missiles.
At the same time, the Kremlin’s forces are expanding their push to the northern border regions of Sumy and Chernihiv, Ukrainian officials say, and Kyiv's outgunned and outnumbered soldiers are struggling to hold them back.
Attacks on civilian infrastructure are increasing, the UN noted, with far-reaching humanitarian consequences, the UN said.
Over 7,500 people have been evacuated from the Kharviv area since Russia's latest cross-border incursion, the most severe yet since the full-scale invasion began.
But many residents remain there, with no access to food, medical care, electricity and gas.
The crisis forced Ukraine's President Volodymyr Zelenskyy to cancel his trip to Spain and Portugal.
Meanwhile, US Secretary of State Antony Blinken sought to rally spirits in Kyiv during his two-day trip.
"You are not alone", he said in a speech on Tuesday. "The United States has been by your side from day one. We are with you today and we will stay by your side until Ukraine's security, its sovereignty, and its ability to choose its own path are guaranteed".
Meanwhile, footage on social media showed Blinken at a club in Kyiv on Tuesday night, where he took to the stage to perform Neil Young's Rockin' in the Free World backed by a local band.
Reactions were divided, as some observers commented that the musical appearance might have come off as tone-deaf, considering Ukraine's struggles in the northeast.
The latest US military aid package is finally on the way after months of political delays. The White House assured it's going to make a real difference on the battlefield.