Fighter jets from the Saudi-led Arab coalition pounded coastal areas south-east of the city on Friday as residents gathered at dawn in an open area for Eid Al Fitr prayers, Reuters reported.
The Iran-aligned Houthis control the capital and most of Yemen's populated areas.
But a Huthi attack some 80 kilometres (50 miles) south of the city provided a diversion, cutting loyalist forces off from the government-held ports of Khokha and Mokha, sparking fresh clashes along the coastline.
Global aid groups cautioned the threat of a major humanitarian catastrophe was growing as fighting drew closer to Hodeida, with the United Nations estimating some 600,000 people live in and around the city.
State-run Saba news agency said the aim of Hadi's first public visit to the country in more than a year was to "supervise" the military operations in Hodeida province.
Fighting In Yemen Port City Threatens Aid Shipments The fighting in a Yemeni port city is reportedly increasing, threatening aid shipments.
Yemeni security officials quoted by AP said that 2,000 troops had crossed the Red Sea from a UAE naval base in Eritrea, with plans to seize the port at Hodeida.
The Security Council discussed the Yemen situation in closed session on Thursday.
Ambassador Obaid Salem al-Zaabi made the comments during a news conference with journalists Thursday.
The Hudaida port is a vital lifeline that is crucial for the flow of food supplies into a country that is on the brink of starvation, as it serves the entry point for 70 percent of the country's imports.
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The insurgents, who have ramped up missile attacks against Saudi Arabia, are often known to claim successful strikes.
Ms Nusseibeh said the offensive to oust the Houthi rebels was "a critical step toward achieving a political solution to this conflict because we know there is no military solution".
The Saudi-led coalition launched an offensive Wednesday aimed at driving Iran-allied Houthi rebels from the port, which is the main entry point for food and aid to the war-torn country.
The council is set to meet behind closed doors to discuss the assault launched on Wednesday on rebel-held Hodeida despite United Nations warnings of a looming catastrophe in Yemen, where millions are on the brink of starvation.
"Four government soldiers were killed in the fighting while more than 10 others were injured", a military source, who preferred not to be named, told Anadolu Agency.
But while the request for U.S. military assistance was denied, the Emirati official said the Trump administration has publicly and privately said it supports the operation, as long as the coalition coordinates with the United Nations, takes efforts to protect civilians and continues to deliver humanitarian aid. The Houthis are backed by Iran. More than 22mn people in Yemen are in need of aid, including 8.4mn who are at risk of starvation, according to the United Nations, which considers Yemen to be the world's worst humanitarian crisis.
The Western-backed alliance says capturing Hudaida would deprive the Houthis of their main source of income and prevent them from bringing in missiles, dozens of which have been fired at Saudi Arabia.
In a press statement after an emergency closed-door meeting, the council expressed "deep concerns about the risks to the humanitarian situation" following the launch of an offensive on Hodeida by the Saudi-led coalition.
The US is seeing growing domestic opposition to its role in the conflict: providing fuel and targeting support to Arab warplanes conducting bombing missions against wedding-goers, among others, in Yemen, though Congress has fallen short of summoning the political will to actually ban the Pentagon and intelligence communities from supporting the Saudi-led effort.
Since March 2015, Saudi Arabia and some of its Arab allies have been carrying out airstrikes against the Houthi Ansarullah movement in Yemen in an attempt to restore power to fugitive former president Abd Rabbuh Mansour Hadi, a close ally of Riyadh.