There is a history of the first recorded flight over the South Pole. This person was American Explorer Richard Byrd and his three companions who made the first ever flight over the South Pole. They flew from their base on the Ross Ice Shelf to the pole and back in 18 hours and 41 minutes.
Richard Byrd was a previous pilot in the first world war, and he had previously been deployed to Greenland in 1924 to help with exploration of the Arctic regions by air. This is what made him so successful in his flight over the antarctic as he had experience with flying over glaciers and sea ice.
Even as an experienced pilot who has faced glaciers before, he still had struggles with the South Pole. Navigational errors due to magnetic compasses not working near the pole, and the most challenging part which was when the plane struggled to gain enough altitude to fly safely over the Polar Plateau. However, they did it and returned home. They had to compensate for navigational errors and difficulties, but the flight was a success. It is possible to fly over Antarctica, it is just not advised due to the dangers that it poses.