I've noticed that the lower or higher your spirits are the lower or higher your amount of forest fights will be.
The only time Spirit affects on whether you gain/lose forest fights is at the beginning of the new day.
Very High Spirit: Gain 2
High Spirit: gain 1
Normal: Gain 0
Low: lose 1
Very Low: Lose 2
That's the only way it affects the game as far as I'm aware of. It might have a small affect on fights but there is a way to gain Very High spirts in a day that you already started with very low spirits or any other spirit level.