The US secretary of state, Mike Pompeo, has embarked on a tour of central European countries aimed at countering the influence of Russia and China, after a decade in which the region has been sidelined by US diplomacy.

“When we’re not here others will show up,” said Pompeo at a joint press conference with the Hungarian foreign minister, Péter Szijjártó, on Monday afternoon in Budapest.

Pompeo was due to dine with Viktor Orbán, the country’s rightwing prime minister, who was shunned under Barack Obama’s administration for his populist style and moves against media freedoms and the rule of law.

The Trump administration has taken a new approach, prioritising engagement over criticism. Orbán’s actions and words on migration, speaking of a threat to “Christian culture” and building a fence along the country’s border, are similar to much of Donald Trump’s rhetoric, and state department officials have also said engagement is necessary to counter the growing influence of Russia and China in the region.

Despite Hungary’s status as an EU and Nato member, Orbán has struck up warm relations with the Russian president, Vladimir Putin, leading many to fear that Russia could be using Budapest as a base for spying and other influence operations in Europe. While Trump has frequently praised Putin, the state department has made it clear that countering Moscow’s influence in Europe remains a priority.

“Hungarians know well from their history that an authoritarian Russia will never be a friend to the freedom and sovereignty of smaller nations,” said Pompeo.

Trump’s friend David Cornstein, an 80-year-old former jewellery magnate, was appointed ambassador to Hungary last year. He has been tasked with building a new relationship with Orbán. Cornstein has said he will not publicly criticise Orbán.

Despite this, Cornstein’s requests to the Hungarian government to keep open the Central European University, an English-language institution founded by George Soros that has come under fire from Orbán, were ignored. The university is gradually moving to Vienna after being refused accreditation by Hungarian authorities. This has led some to question whether engagement was the right approach.

A leaked Hungarian diplomatic cable from December, published by the investigative website direkt36, suggested there was frustration in Washington over Hungarian intransigence on several issues considered as key to US interests.

During his visit, Pompeo made progress on the signing of a new defence cooperation agreement between the US and Hungary, which has long been stalled.

Despite the focus on engagement, Pompeo also met civil society groups in Budapest, including those targeted by the Hungarian government for providing legal assistance to refugees attempting to enter the country. The government has smeared many such groups as “Soros agents”, which are working to destroy Hungary.

On Tuesday, Pompeo will travel to Bratislava, where he will meet Slovakia’s leaders, before visiting Warsaw for a US-sponsored conference on the Middle East. The conference, which has been received warily by the EU, is expected to focus on Iran. Pompeo will briefly visit Brussels and Iceland before returning home.

Pompeo also delivered a message about the dangers of Chinese influence in the region, after a Huawei employee was arrested for spying by Polish authorities last month. The telecommunications company operates across the region.

Pompeo said each nation could make its own decisions but suggested US cooperation could be downgraded in nations that chose to maintain a strong Huawei presence.

“If that equipment is co-located where we have important American systems, it makes it more difficult for us to partner alongside them. We want to make sure we identify the opportunities and the risks with using that equipment. And then they will get to make their decisions,” he said.

Szijjártó complained that claims that Hungary was a friend to Russia and China were an “enormous hypocrisy”, saying links between the countries were no closer than those other western European nations have with Russia and China.