To even begin to understand Kasumi as a character, I'll need to break down the titles of both books. They aren't as simple as I made them out to be in the story, well they weren't supposed to anyway.
I don't understand you.
A phrase that Tsukishima often said to Y/n when she did something he didn't understand. Wrong. I don't understand you when he says that it's more of an I don't understand why I love you even when you do things like this. He doesn't need to understand her to be able to love her, and that's one of the biggest misconceptions in the story. A big reason Y/n feels the way she does, she doesn't realize how much he loves her.
When he says "I don't understand you" he's saying he loves her, loves her enough to stay even though he doesn't fully understand his own feelings. It's showing that he thinks she is worth the confusion and struggle because he loves her.
Just let me understand you
Knowing what "I don't understand you" means now, this should be easier. Throughout the second book, it's not a matter of him wanting Y/n to let him understand her, he really means "just let me love you". When Y/n says that she doesn't even understand herself, she doesn't love herself. Or she doesn't quite understand why anyone would ever love her. In her mind she doesn't deserve it, in turn, she refuses to accept it.
Over the course of three years, Tsukishima finally learned and understood his feelings for Y/n fully, with the help of his supports. Supports Y/n doesn't necessarily have/accept, a representation when Y/n cried alone in her room, while Tsukishima had his mother. Yes, they suffered similar pain but did not have the same long-lasting effects.
When Tsukishima says it's only been three years he's referencing the six-year deadline promise he made to ask her to marry her. Not that she remembers, but that's the point. It shows that he's never stopped thinking about her, and he's not going to even try until he keeps his promises to her, even if she doesn't marry him in the end.
Now that Tsukishima has understood himself and his feelings, all that's left is for Y/n to do the same. Not understanding her feelings, but understanding the feelings he feels towards her, and realizing just how deep and strong they are. Bring us to Kasumi's existence.
Kasumi-meaning mist. The mist Y/n must clear to finally be able to see and understand.
Kasumi plays a crucial part in the Tsukishima path, without her Y/n and Tsukishima would never even have a chance of getting back together. It's clear the Kasumi is much like Y/n before she let herself feel completely worthless. She has no biological parents, as they are both dead, Y/n's biological parents left.
She has issues with abandonment and struggles to comprehend why people need to leave her. She bonded to the photograph of Y/n, as it was the singular thing her mother had left her. She has only two photos of her mother, one with Y/n and another with her and her mom. Her giving the two photos to Y/n show the bond that she already created in her mind, the trust she's leaving with Y/n. The promise Y/n made to come back and see her.
Y/n will eventually see that Kasumi is very much like herself, and she almost stands in like Tsukishima. How he was for her. She'll come to see that the love he had wasn't just romantic, he cared about her, wanted her to be safe, and nothing brought him more joy than seeing her smile. Y/n will come to feel the same way towards Kasumi.
Eventually, she will understand what Tsukishima was going through when she left, she wasn't helping him, she was causing more harm. She doesn't want to let the thoughts of doubts cloud Kasumi's mind. She doesn't want Kasumi to end up like her, so she wants to give her the love she did not receive as a child.
Y/n will come to learn that she is needed and understand that Tsukishima not only loved her but needed her as she needed him. She will come to understand everything, then it will get all better. But not without one more loss.
And there we have it, my absolute genius.
-Crouton