As a side note, I have the impression that the elf girl starts to miss a lot after you give her 2-3 bottles of Wine in a row, be it Normal or Elven.
The MC at some point is able to hit the 5 bottles without the Agility Potion, put I'm not sure if it is because you played the Archery Game a few times or because you have gained X levels; or a combination of both circumstances.
The higher your Agility the less likely you miss on each shot, she tells you archery requires good Agility. But even good Agility can miss sometimes.

Agility goes up with level (not as fast as other Attributes, but still). Wine debuffs Agility and Elven Wine debuffs twice as much, click on them from the Item menu and it shows you that. So the more she chugs the drunker and less Agile she gets.
Is more efficient to choose sexual favors first, then you can exchange Elven Wine for Sacred Water
I get 3 Elven Wine and 13 regular for the first time I do the contest, give her the Elven Wine on the first three rounds, and go for the lewd acts on the first three wins before getting water on wins. By the end she's so smashed I consistently complete the lewd acts and also fill up on water (ten wins = 99 water) before I run out of wine to give her. After that you can use regular wine when you go for water refills (or more banging), and have probably leveled up to where your own Agility almost never misses anyway. Remember to pour out the wine she gives you or you'll be smashed as well!
14 coins
TIP: Holding down a mouse button speeds through dialogue (everywhere, not just with Liandra) & cutscenes (like the archery contest), stopping when a choice menu pops up. Holding down ESC/ENT also speeds through but automatically picks a choice for you if a choice menu pops up, which may not be the one you want. I use a mouse pretty much exclusively.