Look carefully on Jack's jaw in these pictures, especially the left side of the picture. Under the 4th eye, to the left from his moustache there is a red ellipse. The picture is from the Dead Man's Chest (found here).
Now look at this picture (found here) from The Curse of the Black Pearl and see the difference between the pictures.
Any ideas when he got the scar? I don't think it's a scar from a fight as it looks fresh even in the latest movie (Dead Men Tell no Tales/Salazar's Revenge). It looks like the skin was ripped off in some strange way. Similar to the look of a wound after a fat/pus surgery.
Also, it's just the character's scar, not Johnny's. See this calendar for reference.