No, prefixing a h is not lenition, it's Irish's third mutation known as pre-aspiration.
a = his, lenites.
a = her, pre-aspirates.
a = their, eclipses.
Although lenition is indicated in writing with a 'h', this doesn't really relate to the pronunciation. 'dh' is a completely different letter to d.
The pre-aspiration cased by 'a = her' however is a genuine h sound being added to the word.
I see what you mean. You mean when the sentence comes with the drop down menu to fill in the last blank. I just looked at the six solutions, and every single one would be grammatically correct, but would require different meanings of "a." But without a translation provided, there's no way to know if Duo wants to have it mean "his", "her" or "their." I sent an error report, and make sure you do, too.