Every different colour soul in Undertale represents a different emotional theme, as well as having their own gamemode. My theory is if this applies to monster souls as well
Apart from the orange soul of bravery, we've seen every soul in action in Undertale, and how they relate to their themes, I.e. the yellow soul has a gun, akin to how a sheriff may use one for the sake of justice. Therefore, there should also be an emotional theme and gamemode attached to the white monster soul, which isn't just the same as determination.
I believe that the white monster soul represents the emotional theme of freedom, given how most monsters will talk about how they long for the freedom of their race.
However, the white monster soul should also have their own respective gamemode. That's why I think that it should look like the chase sequences in the game, but with the ability for the player to fight back with their own magic. Of course, the fight would be stuck to one screen, instead of moving across multiple ones.