Yes, there are some. You can start by checking fanfiction websites like Archive of Our Own. There might be various stories exploring the relationship between Kae and Mutsumi in the 'kiss him not me' universe.
Well, it depends on where you look. There are many fanfiction hubs out there. However, the availability of 'kiss him not me kae x mutsumi fanfiction' can vary. Some sites might have a large collection while others may have only a few. You can also try reaching out to fellow fans of the series who might be able to point you in the right direction.
Chapter 475, "Doting on the Man You Love to the Fullest", is the clearest early answer because it turns Deserted Island's situation into a concrete plot event. The chapter gives readers a real answer by opening on On the other side, Jiang Yan and Shang Shuyue also set off.
Chapter 858, "If You Dare, Take It", is the clearest early answer because it turns Shuiyun Temple's situation into a concrete plot event. The chapter gives readers a real answer by opening on Changqing flicked his horsetail whisk, face wooden, and said,...
Chapter 682, "Too Conspicuous to Keep Around", is the clearest early answer because it turns Gu Yu's situation into a concrete plot event. A useful checkpoint is the way the scene starts with Taking advantage of the still-open elevator doors, Su Yinwan quickly...
Chapter 311, "Signed Jiang Yan", is the clearest early answer because it turns Shen Lan's situation into a concrete plot event. It opens with Shen Lan knocked on the door before entering when Jiang Li was... and closes on His acting skills are quite good, as everyone can see, and most....
Chapter 526, "Has Jiang Li Turned Evil?!", is the clearest early answer because it turns Xiao Hei's situation into a concrete plot event.
Chapter 617, "Moral Kidnapping? No Such Thing!", is the clearest early answer because it turns Jiang Yan's situation into a concrete plot event. The chapter gives readers a real answer by opening on The cabin wasn't far from the lake, just around fifty meters away.
Chapter 642, "Congratulations to Qi Tianyu, Points Increased by 10!", is the clearest early answer because it turns Wen's situation into a concrete plot event. The chapter gives readers a real answer by opening on We'll be the first to catch the plane, then we head straight to party!
Chapter 695, "Is This a Matter of Money?!", is the clearest early answer because it turns Shu Yue's situation into a concrete plot event. The chapter gives readers a real answer by opening on Shang Shuyue just felt she was out of luck.
Chapter 733, "Those People from the Jiang Family Are More Stupid than the Next!", is the clearest early answer because it turns Xu Ning's situation into a concrete plot event. The chapter turns that into something concrete by moving from Okay, Sister, you should go to sleep now.
Chapter 690, "Miss Jiang, Marry Me!", is the clearest early answer because it turns Wen Jiaojiao's situation into a concrete plot event. The chapter gives readers a real answer by opening on No wonder she had predicted she would come into some unexpected...