We cannot grasp 4 spatial dimensions. It is not even known if they exist. It is proven that in 4 dimensional space a planetary system could not be stable. So for all intents and purposes, ouside mathematics, there exist only 3 dimensions. Also if there were 4 dimensions, why should objects only have 3. If you count time as a dimension, objects sure have a length in those as well.
Per definition an added dimension is a direction perpendicular to the others. It is easiest to imagine as moving in time (althoug that is not a spatial dimension). So if you could move along the non spatial axis to a point where the secret cave entrance is open, you can then use your regular 3d movment to go in and collect the secret.
What about direct 4d movment then. If nothing exists perpendicular that would block your path, you could initiate a movement and while moving shift along the perpendicular axis for a short time to arrive behind the barrier. If nothing there blocks your path, you cannot just shift there and move then, because even the ground you are standing on would not exist, but neither would gravity. Or Air.
A movement ability or device that would just transport you from A to B would be indistinguishable from plain old teleportation, maybe by how long the movement would take. That has been done countless times. Calling it 4d movement adds nothing but the can of worms of side effects. (including people mistaking it for time travel). Like dying from lack of oxygen if your travel takes too long. Travling along a new axis does not make the distance shorter. Yes, yes, in higher dimensions the space could be curved and the distance could be shorter, as explained in those scify stories with wormholes. But why should it be curved just so that your travel through a mountain to reach a secret cave is short. Your ability might do that, but this would be like those secondary super powers. Just call it teleportation. Phased movement. Shadow Step. Ghost realm. Hyperspace, etc.