Alicorn is the most common word used to describe said beast, however, in the Xanth series, there are also PegaCorns and UniPegs. All three creatures look exactly the same in the series, which is a subtle joke, and the only way to tell them apart is to ask them.
This lists references citations where Alicorn is used to denote a winged unicorn. I can find no such list for any other word that could mean the same thing. Note that the first use of the word according to that is by Piers Anthony, the author of the Xanth series I note above. He may have have coined the use of Alicorn as a flying unicorn in 1984, in his book "Bearing an Hourglass", but there are sources from earlier than that which imply that he di not, see below. I'm going to research this further, as I distinctly remember reading about Alicorns in another of Anthony;s books, one of the Xanth books as I note above, prior to reading the series that contains "Bearing an Hourglass". When I find the canonical reference I will edit this and include the title and page number of the first place I see it in the Xanth series as well.
This text, from 1930, clearly lists an Alicorn as a seperate creature from a Unicorn, along with a Monocerous and a Rhinocerous, and suggests that all four beasts horns have magical properties.
Here is another text with the use of an Alicorn as it's own beast as well, although not descriptive, it says that an Alicorn has a curved horn, and it's from 1930.
Interestingly enough, as an aside, here is a text that cites Alicorn as being a real substance, most likely being the horns of Narwhals that were found and attributed to having belonged to a Unicorn. Here is a story in which the plot revolves around Unicorns being chased by the antagonist into the ocean, perhaps the author is suggesting that's what happened to Unicorns in reality.