Damage or undefeated monster power first?

[Sliska Zafir]

I recently encountered a Giant Fly, failed, took damage and had to bury a card from my discard.

In this case, does damage happen first, or after the bury effect? This makes a difference because the damage could be buried (which btw, had my healing card in it).

[MorkXII]

If you took damage because you failed the check to defeat, then you would discard cards for damage before you bury cards for failing to defeat the fly.

Taking damage for failing a check is part of the "Attempting the check" step, and then the card bury takes place in the "Resolve the Encounter" step (pg 9-10, Core rule book)