You have finished the show by now but if you are still confused about the beginning of episode 13 ill quickly answer the q's you had. He called his henchman mi-jeong because if you remember earlier in the episode the random guy at the bar asked him why Gu always calls him(the henchman) random names and he replied its cus he doesnt like his new name. So thats what happened there but this time Gu used mijeongs name to call him cus he was thinking of her and missed her. Then he wanted to do something for the henchman because he was thinking of her so the henchman said he wanted to go back to his hometown and thats what that was about, nothing more.
Also the whole drama till now takes place in 2019 we jump to I assume months since episode 12 ending starting in episode 13 with Mr gu then it kinda jumps back and forth with the end of episode 14 being the present but thats why so much changed because it has been months/years but I think you will know that watching ep 15,16.
Anyways hope this helped a bit and didnt confuse you more haha. This show has a lot of symbolism so even the most random scenes(baby, car chase with the dad) look useless or random its not theres a lot of thought behind these (esp the baby scene) if you think deep enough. I recommend reading online if you are truly curious because this show is so brilliant but if you take things at face value it might not seem as good.