Given that "amharc" begins with a vowel, how could it be indicitated, if it was necessary to do so, that it was his or her or their sight which was in question?
The problem for me is more about what you do when a noun begins with H. A hata is open to interpretation. What do you do if it's critical to say his, her or their and it's not clear from context?
Words that begin with "l", "n" or "r", and words that begin with "sc", "sp", "st", "sm", "sf", and words that begin with the "loan letters" of "h", "j", "v", "w", "z" are not lenited (a least in writing - apparently "l" can be lenited in speech in some dialects).
If it's critical to say his or her, then you add the necessary context, perhaps by mentioning the owners name.