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Myung-Ha is not 'spineless' he is a poor man being exploited by the healthcare system. He doesn't have the economic privlege of upholding the mores of romantic entanglements in the face of financial stability, especially when his dying mother is being used as a bargaining chip by some psycho bitch who pissed down his throat to scare his boyfriend away. Literally.
A central theme of this manhwa has been the lengths people go to for money in a society that not only places more value on the image of economic stablity than it does on actual human welfare, but demands you maintain an air of decency too. Whoring yourself out isn't just running your name through the mud, it's social suicide. And he was doing it for his mother.
All of these people are desperate for money. Why is it such a surprise to everyone that Myung-Ha would accede to her demands? There's other avenues but he's never going to find one with less bumps in the road.
Jung-Won may be hearbroken but he knows what it's like to be that desperate. He know's Myung-Ha doesn't love her either and he isn't going to just let Myung-Ha throw his life away like that. There's obviously risidual feelings.