The battle for Yemen's Red Sea coast moved into the mountains on Tuesday.

Forces loyal to Yemen's Saudi-backed government launched a large operation to take al-Waziiya, a mountain district in Taiz province, Al Jazeera reported. It sits between the city of Taiz and the port of Mokha. The army said its troops took hills overlooking the al-Mansoura area as Houthi fighters pulled back. The Houthis denied it and said they pushed the attack back.

Those hills matter. Al Jazeera's reporter in Sanaa said the government cannot hold Mokha without first taking al-Waziiya. The Houthis seized Mokha and land near the Bab el-Mandeb Strait in a fast push in early September.

The government said Monday that it had already retaken Mokha. CNBC said it could not confirm that claim. The Houthis deny losing the city. Mokha lies about 46 miles north of the strait. It once gave its name to the mocha drink.

The Houthis also struck back. Yemen's government said they fired at least two ballistic missiles and several drones at Aden's airport. One missile landed near the runway minutes before a flight from Cairo was due to land, the transport ministry said. The plane was sent to Jeddah instead. The Houthis said they were aiming at Saudi military supplies near the airport. Saudi Arabia said it shot down a missile north of Riyadh.

The Houthis also claimed attacks on Saudi Arabia. Attacks on Abha International Airport and King Khalid International Airport in Riyadh on Tuesday and Wednesday killed three people and hurt 36, Saudi Arabia's General Authority of Civil Aviation said, Al Jazeera and The Straits Times reported. At Abha, two women who lived in the kingdom, one from Morocco and one from Algeria, were killed and 28 people were hurt. In Riyadh, a Sudanese resident was killed and eight people were hurt, five of them Saudis. The U.S. Embassy in Riyadh told Americans in Saudi Arabia to stay alert.

The fight has widened. Saudi Arabia says 100 of its fighter jets are backing the offensive, Euronews reported. Pakistan and Turkey agreed this week to send forces to Saudi Arabia.

The Bab el-Mandeb carries about 12% of world trade, Euronews said. It matters even more now that Iran has mostly closed the Strait of Hormuz.

Yemen's war began in 2014, when the Houthis took the capital, Sanaa. The latest round of fighting has killed at least 959 people and forced more than 174,000 from their homes, according to UN agencies.

What we know: Heavy fighting is underway around Mokha, and attacks on Saudi airports have killed three people.
What we don't: Who truly controls Mokha today.