One synonym for the phrase "hair dog that bit one" is "troublesome situation." This phrase is often used to describe a difficult or challenging circumstance that has caused distress or discomfort. Another similar phrase that could be used as a synonym is "thorn in one's side," which refers to a persistent problem or annoyance that is difficult to get rid of. Additionally, the phrase "burden on one's shoulders" can be used as a synonym, as it indicates a heavy weight that one must bear and cannot easily be shaken off. All of these phrases suggest a feeling of frustration and difficulty in dealing with a persistent problem or challenge.