No, Kana doesn't mean that. Kana was afraid that if she was too spoiled and relied on Damian, it would make the man bored and feel tired of her. Kana is afraid of being thrown out. Unknowingly, Kana started to sob and buried her face in her hands until the blood from her wound started seeping through the cotton that Damian put on.
Kana wept bitterly—why did everything not go smoothly like this? Doesn't she deserve to be happy? Was Damian disappointed in her now? What if Damian dumped her, and she now loved that man? Kana didn't know how to live without Damian by her side. Apparently, without Kana realizing it, Damian had the most important and valuable role.