Do I use Togekiss' "Precious Gift" Ability before or after taking prize cards at the end of my turn?
Precious Gift is used at the end of your turn after attacking and checking for a KO but before Pokemon Checkup. So if your attack causes a KO you would take Prize cards first, and then you can use Precious Gift.
Source: Obsidian Flames FAQ Addendum; TPCi Rules Team (2023-08-10)

Relevant cards

Note: All applicable cards may not be listed below.
SV3_85
SV3_85