There has been heated argument about this on the Welsh course. It probably varies with where you are from and how old you are. I think in general that has toothache is more modern and has a/the toothache is older, but arguing about it won't make the toothache go away.
I know that "Ian has the toothache is marked correct. That's the usual Scots way of saying it. Were you marked wrong for "a"? Maybe it's because the Gaelic includes the definite article so if you put an English indefinite article in then it's incorrect?