What does pushing her in the lake achieve?????

its to remind her place, it means that she will soon entered the palace as a concubine and to make sure that her (fl) influence will be lesser than the empress. the soon to be empress obviously knew her fiance's obssession it can risk her place as a empress she definitely wanted the influence and power of being a empress and she think that if she get the fl as her lackey/subordinate she can secured her position.

Are u sure? But she helped her right back up? And she’s trying to make a deal with her she doesn’t look or sound threatening at all..