Hey Dave,
you can enter G-code commands into the command entry field of the MDI Tab (in the Tabbed Section). The machine will do what you want, commands one by one.
By the way, an excellent way to familiarize with the g-code commands by trying out what they do… 
I agree with BJ that in cases of power outage you should avoid to move the axes or to home the machine.
To resume at a point in a program is something the Onefinity Controller does not offer (more professional CNC Controllers do), but you can achieve this manually. You got to prepare a program that only has the “rest” of the program (know at which line in the program the machine was), but you must keep the modal settings at the beginning of your program or code block.
If I understand you right, you wrote down the positions. Did you write down the positions, absolute positions, and offsets for all axes?
What do you mean with “I left the laser at x,y zero but I also wrote down the absolute coordinates”, do you mean, the laser was at “workpiece origin”? And you know where this was because you have written down this position in machine coordinates? If so, it is easy to restore the situation.
Yes, after homing you first move the laser to the coordinates you want by entering the appropriate G-code ‘move’ command into the command entry field of the MDI Tab. E.g. if your position of the laser was a X at 150 mm and Y at 100 mm, and you are at home position, you enter ‘G0 X150 Y100’. G0 is “rapid move”, a laser will not fire during such a move. The Laser will then move to this position.
Then to set the workpiece coordinate system to be centered at this position you use the ‘G92 axes’ command. It sets the current position as the new workpiece zero. G92 is the command that is usually used after probing. For example if you probe Z with the touch probe, a ‘G92 Z15.4’ command is used because 15.4 mm is the thickness of the touch probe which is substracted this way. ‘G92 X0 Y0’ now will set the current position as the new workpiece zero (workpiece origin) for X and Y.
You could also leave the laser at machine origin (“homed”) and instead enter ‘G92 X-150 Y-100’, and then move to that workpiece origin by entering ‘G0 X0 Y0’.