If I understand your explanations in the comments correctly (you simply copied values from the camera to the plane and moved it away on the camera's Z-axis) then the easiest way would have been the following:
Let's say you want a distance of 5 m from the camera. While the plane was in the same location as the camera and rotated the same way, to select the plane and just press GZZ-5Return, this would have moved the plane -5 m away in Z direction (away in front of the camera is -Z).
But now that the plane is already away from the camera (but still has the correct rotation I guess from your explanations?): are the objects parented to the plane? If not, select the objects first, then the plane and press Ctrl+P > Set Parent To > Object.
Then select the camera, Shift+S > Cursor to Selected. Now select the plane, Shift+S > Selection to Cursor. After that you have the initial state of the plane being in the same location as the camera and having the same rotation, just like before you can now move it with GZZ-5Return.
If you want a better solution where you can easily change the distance afterwards simply by entering a Z value for the plane, you can change the steps a little bit after you moved the plane to the camera's location:
First select the plane, then Shift select the camera and now parent the plane to the camera, but not just with Object like before, press Ctrl+P and choose Object (Keep Transform Without Inverse).
If you now select the plane and look at its Location and Rotation values, they are all 0 - set relative to the parent object. Well, sometimes there are slight inaccuracies like -0 or 0.00001 etc., but you can manually set them to 0 then.
To bring the plane now into a specific distance from the camera like e.g. 5 m, just enter -5 m into the location's Z value and change it whenever you want a different distance.