So this means the GAIDEN -wooley eyes- is kind of like an expansion or side story to the main story of - a Tale of Cutthroat Lovers-. Gaidens usually come out a year or so after the main story due to fan demand for more character development. Sometimes to tie up loose ends, to give closure to certain characters or even What-If scenarios. Famous visual novel gaidens I can think of are Fate/Hollow Ataraxia for Fate/Stay Night, Dive X for Baldr Sky, Sumaga Special for the original Sumaga, etc. You are free to play the Side Story first and then the main story. Each has it’s own kind of impact as usually the Gaidens have more ideal outcomes for characters compared to the main story, possibly based on fan response. Muv Luv Extra and Alternative spawned a ton of Gaidens, as did franchises like Koihime Musou and, of course, Fate/Stay Night.

A little bit of background about Cutthroat Lovers. Originally released in 2017, probably from the same people who came up with the infamous Euphoria, a 2011 title. Strongly recommend reading reviews of Euphoria before getting into this one as Euphoria seems to be as dark as stories can get (I hope). Cutthroat Lovers is categorized by Clockup as a romance novel though, while Euphoria is described as “Immoral Hardcore”. Still, i am treading carefully with this one as it has filter options for scat and gore. Seriously, turn these on if you are new to this kind of thing. I wish Muv Luv Alternative had this when it first came out. Do not go into this game looking to be titillated. You will likely be kicked in the teeth for it. It feels like a dark, violent underworld crime drama similar to classics like Crying Freeman, Phantom of Inferno, Noir, etc. Think “The Girl with the Dragon Tattoo” but with explicit scenes, both tender between the main characters and horrific with the villains. Expect to be railroaded into some horrific bad endings from seemingly reasonable choices that give insight into the dark sides of some of the characters and situations that prepare you for alternative plot development. I cannot say if it will reach the heights of Nitro Plus novels and am a little terrified of how much Euphoria type horror may be found. It is really not for everyone so stay away if you are not keen on dark adult fiction with extreme violence and criminal underworld immorality.
(FYI, I would not go near Euphoria even with a hundred foot pole. Just give me a comedy like Subverse. I do hope Sacrifice Villains will be comedic.)