Showing that her closeness to sing will never change because he’s the only one who helps her and understands her and she considers him above the emperor. To remind us of that and give us some fluff before going back into action. She’s not really being weird. You never noticed she always goes to sing and her dogs first? The only time she goes to the emperor first is if she wants to update him. Her life is dedicated to her puppies and then her consorts and then her husband. This is probably the moment before that shifts and the emperor becomes her number one.
This is literally the perfect way to place Sing in this manhwa. Hes not "the one devoted to her only wanting to fuck her in the end" like the usual second male lead. He wasn't in the love interest. Hes just there serving FL with no romantic feeling. Literally clear conscience that FL doesnt need to chose one person to stand by her. Imagine if Sing was a love interest and we got "Actually I was The Real One" type of shit. Hell na.
Why is the Fl being so Weird. Why is she suddenly asking Sing to bathe her?? What did that scene contribute to the story?