Oddly, the accounts I have seen say the wound was on the left side of the face, next to the nose. Perhaps they mean the left side as the surgeon viewed it?

Anyway, the wound was caused by an arrow that pierced the teenaged Prince Henry's face at the battle of Shrewsbury, 1403, and penetrated to a depth of six inches.

It was extracted by a surgeon named John Bradmore who had a special extractor made specifically for the job. His account is quoted in a pdf from Glasgow University and was dramatised by Discovery Channel.

First, I made small probes from the pith of an elder, well dried and well stitched in purified linen [made to] the length of the wound. These probes were infused with rose honey. And after that, I made larger and longer probes, and so I continued to always enlarge these probes until I had the width and depth of the wound as I wished it. And after the wound was as enlarged and deep enough so that, by my reckoning, the probes reached the bottom of the wound, I prepared anew.some little tongs..( he describes the extractor)

I put these tongs in at an angle in the same way as the arrow had first entered, then placed the screw in the centre and finally the tongs entered the socket of the arrowhead. Then, by moving it to and fro, little by little (with the help of God) I extracted the arrowhead. Many gentlemen and servants of the aforesaid prince were standing by and all gave thanks to God.

Now that certainly sounds like it would have left a scar!