Ok, so if a card or ability is "once per turn" if you have not used it yet by the mythos phase, you can go ahead and use it during the mythos phase and then at the beginning of investigator phase, it would reset?
You will notice that not a single card says "once per turn". They will say "once per round" instead. A round is specifically defined as one investigator phase and one successive mythos phase. Which means that, indeed, when the item wasn't used in the investigator phase, it is available to be used during the mythos phase.