Donald Trump has met Pope Francis at the Vatican on the third leg of his first overseas trip as US president, one year on from a public clash about what it means to be a Christian.

The hastily arranged get-together featured a discussion about healthcare and immigration, and a light-hearted exchange between the pontiff and Melania Trump about whether she fed her husband a popular type of Slovenian cake.

The opening exchanges between the pope and the president – who disagree on issues ranging from the need to fight global warming to welcoming and integrating migrants of all faiths – began on an unusually sombre note, with the pope not exuding his usual warmth and cheerfulness.

Francis also signalled to the president that it was best to talk out of the earshot of reporters and cameras, who were quickly led out of the papal library where the two had a 29-minute chat with only a translator present. Trump could be heard saying it was an honour to meet the pope.