By Barbara Erling

WARSAW (Reuters) - Poles vote in a closely fought presidential election run-off on Sunday that pits the centrist, pro-European ruling party's candidate against a nationalist challenger who draws inspiration from U.S. President Donald Trump.

Opinion polls show Warsaw Mayor Rafal Trzaskowski of the ruling Civic Coalition (KO) narrowly leading Karol Nawrocki, the candidate backed by the main opposition Law and Justice (PiS) party, ahead of the June 1 second round, which will have consequences for EU ties and relations with Ukraine.

The two candidates embody contrasting visions of Poland's international relations, security strategy and social policies.

Trzaskowski, 53, the Oxford-educated son of a jazz musician, champions a liberal agenda that stresses the importance of women's rights and strong ties with the European Union and NATO.

Nawrocki, 42, a historian who has been endorsed by the Trump administration, has positioned himself as a newcomer with no political baggage. He wants Poland to follow a path inspired by Trump and regards Washington as Warsaw's key ally, not Brussels.

Julia Szwoch, a 25-year-old student of pedagogy from the northern port city of Gdansk, said she would back Trzaskowski in Sunday's run-off vote.

"I want to live in a Poland that is inclusive and takes into account the needs of all people," she said.

In contrast, Michal Diug, 21, an electrotechnology student, said he would probably back Nawrocki "so there isn't a monopoly of power" at the top of the Polish state.

Trzaskowski has promised "peaceful cooperation" with the government of Prime Minister Donald Tusk and to help undo changes to Poland's judiciary under PiS that critics in the EU and at home said had eroded democratic checks and balances.

In the October 2023 election, PiS lost its parliamentary majority to Tusk's coalition after eight years in power, amid widespread public dissatisfaction over PiS's record on women's and minority rights.

However, Poland's outgoing president, Andrzej Duda, a PiS ally, has used his veto power to block many of Tusk's reforms and Nawrocki vows to do the same, describing the election as a referendum on the government.

"If my opponent, Karol Nawrocki, wins there will be only chaos," Trzaskowski told a mass rally in Warsaw last Sunday.

Trzaskowski faces a tricky balancing act ahead of the second round. He wants to appeal both to supporters of the far-right Confederation, whose candidate placed third in the first round, and to progressives disappointed with Tusk's government.

He has said he is against accepting more refugees after Poland took in almost a million Ukrainians fleeing the war next door, but remains committed to liberalising abortion laws.

"We have to go wide," said Dorota Loboda, a Trzaskowski campaign team member.

CONSERVATIVE VOICE

Nawrocki said that after years of what he calls "mistakes", including energy dependence on Russia and the EU climate pact, Europe needs "a strong, conservative voice, also from Poland".

"I will make sure that Poland becomes the leader of the European Union in transatlantic relations. We have the best cards for this," he told the Conservative Political Action Conference in Poland on Tuesday.

His campaign has been characterized by nationalist rhetoric and a focus on traditional values. Nawrocki opposes abortion and aims to keep coal mines operating until Poland develops its own nuclear energy. He opposes EU migration policies.

Although Nawrocki says he favours continued Polish support for Ukraine's war effort, he says he would not ratify any application by Kyiv to join NATO. Trzaskowski argues that Poland's own safety depends on Ukraine joining NATO.

Nawrocki has faced allegations about his past in liberal media outlets, including a report that he lied about owning a second property and that he acquired it from an elderly man in return for a promise of care which he did not provide. He has denied any wrongdoing.

Trzaskowski has faced questions over whether a series of advertisements criticising his opponents subverted campaign spending rules. He also denies any wrongdoing.

The election takes place against a backdrop of rising anti-establishment sentiment in Poland, with the far-right achieving their strongest ever showing. Pollsters expect far-right votes to go mainly to Nawrocki.

"We can already see in these elections that these anti-establishment emotions are driven by a perception that the politicians have again failed to deliver," said Filip Pazderski, an analyst at the Stefan Batory Foundation.

"The situation will become ever more difficult for the current government," said Pazderski, adding that if Tusk fails to deliver on his promises, existing tensions between coalition partners could intensify and a snap parliamentary election might become inevitable.

(Reporting by Barbara Erling; Editing by Gareth Jones)