Looks like I cleaned where sappa's daoka core was stored before even entering sappa. Now I try to complete the quest to open the Daoka and it tells me where the daoka is, but I do not have the core in my inventory. What can I do to complete the quest?

The quest tells me to go to the Daoka and input the core, but I can't as seen in the image

I've done that plenty of times with various ones, including most probably Sappa, and haven't seen any problem (played through that part of the game at least 3x now). So, am not sure what is going on here. Might be something very specific that you did differently than me.

Could try restarting the game, pressing the use key anyway to insert a core, or loading an older save if you have one. You would need that (its required) so no point in continuing until you solve it.