Assuming a character or creature with very little speed - say, 15 feet. It gets hit with a ray of frost, reducing its speed to 5 feet. It is attempting to move through difficult terrain. Can the creature in question move through the difficult terrain without dashing?
Yes, normally - but not when using the Playing on a Grid variant rules
The basic rules say of difficult terrain:
Every foot of movement in difficult terrain costs 1 extra foot.
The default presumption of the rules is not that you are playing using a combat grid. If a creature can only move two and a half feet in one turn, they still move two and a half feet; they don't have to snap to an arbitrary grid, they can still make progress moving.
However, the Playing on a Grid variant rules state:
Rather than moving foot by foot, move square by square on the grid. This means you use your speed in 5-foot segments. This is particularly easy if you translate your speed into squares by dividing the speed by 5.
If a square costs extra movement, as a square of difficult terrain does, you must have enough movement left to pay for entering it. For example, you must have at least 2 squares of movement left to enter a square of difficult terrain.
Under these rules, a creature with only one square of movement available (because it has a movement speed of only 5ft) cannot move into a square of difficult terrain unless it Dashes (or otherwise gains extra movement), because it must have 2 squares of movement available to enter the space.
As a DM, I would probably let a creature in such circumstances move one square every other round rather than forcing them to use an action to Dash in order to make any progress. They're still considerably slowed, but they don't suffer any extra penalty compared to the default case of not using a grid.