[@CollectorOfMyst]The most reliable way to recover determination is rolling a 1 for movement, yes.

Some items help with determination (for example, Hawkeye Harold's personal item, but they are also in the gear/artefact decks I believe) as well as some encounter cards and some bad cards (darkness and maybe some dread cards as well, I would need to check to confirm though)

Determination allows you to reroll a dice. It could be used defensively to reroll failed defense dice, offensively to reroll missed shots or combat damage, to reroll scavenge checks or reroll movement, or reroll a skill challenge (such as the "Cunning" Test Dan did earlier, for example, or the "Luck" test Pastor Jimmy did recently) off the top of my head. There might be a few more examples, like dodging environmental dangers, but I think those are the main ones.

Scavenge checks give you 3 dice to roll, and a 6 succeeds and you find a "scavenge" card. If you fail some of the dice (for example, roll 1, 1, 6 on your scavenge check) you can spend determination to reroll the failures (perhaps ending up with something like 6, 5, 6)