Indeed, かぇってきてね is obviously derived from the verb かえってくる, which translates to "come back, return". Either she's telling him "Hurry up and come back (later)" because she wants more of what they did (and other stuff, probably), or "Hurry up and go back (to your home)" because she doesn't want him to get into trouble with anyone who might get suspicious if they notice he's not at home. In the latter scenario, I'm assuming he's not married, because I don't believe Himura Kiseki would risk portraying adultery as sympathetic unless the wife is an unsympathetic frigid bitch, which would be an unusually negative portrayal for a female character in his works.
The fact that it's peeling in the corner makes me think it's an object rather than just a bad censor job. But it still looks unfinished/unidentifiable. Possible specific thing from an episode?
The scene is about the 2:40 mark from episode five. I'm not seeing an object that would match up.
Apparently, this pose and others like it reference the ED to a JDrama adaptation of some manga. https://vimeo.com/187538552 (from the one-minute mark onward).
