Princess Asagao is one of Genji's cousins whom he corresponds with off and on. He's also somewhat romantically interested in her, though she cuts off contact with him after Suzaku becomes emperor. Genji is unable to ever woo her effectively, as she takes a priestess position that takes her far away from Genji's reach.
Princess Asagao Character Timeline in The Tale of Genji
The timeline below shows where the character Princess Asagao appears in The Tale of Genji. The colored dots and icons indicate which themes are associated with that appearance.
...(effectively making his neglect her fault). Fearing a similar fate, one of Genji's regular correspondents, Princess Asagao , preemptively cuts off communication with him. Genji's in-laws are also upset with him, though... (full context)
The Sacred Tree
...dedicates himself to Yugiri's education and neglects his many ladies. He's sad, however, when the Princess Asagao is called to take a priestess position. Regardless, he continues to write to her. Suzaku... (full context)