I'm confused, with the evidence and also condition that are shown in this chapter the male lead could easily win the child custody at the moment, since his wife are cheating and he also have some strong evidence too (those photos of the wife's chat history and also the wife meeting her cheating partner). As long as the male lead hire a decent lawyer he could win the child custody at the court, and if the evidence are not enough he could also stated that the wife could not take care of the child properly, since she does not do any decent housework and also always come home late(?). But, well since it's still chapter 1 maybe there's some plot twist and other reasons why the male lead would not divorce his wife at this time.
Overall the premise are interesting, but somehow the translation are a little bit bad (but still understandable). Anyway thanks for scanlating this chapter, it was a good read.