Hey Mike,
yes, if after a power outage you know the offsets that were in the offsets column before it happened. What I mean, you won’t have to re-probe again, because you tell the machine where its workpiece zero was, and in the first case described below, additionally where it is since the power failed (this case works only if the power outage happened when the machine was paused).
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First I wrote, you could take a photo of the positions and offsets on the control page (more than one people reported to do that before every run after initial workpiece probing, especially if their circuit breaker tends to trip).
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But then I found out that when you probe, the log file records the offsets and you can find these values later again in the log file, even after a power outage or a reboot. This means you don’t really need to make a photo, at least not for restoring the offsets.
As an example, let’s say, you probed your workpiece and started to run the program, and then you press pause. Let’s say the values are this way then, and you made a screenshot of it:
In this example we assume that the machine was not moving the axes in the moment of the power outage, but paused as you said. Now you get the power outage, and after reboot the user interface looks this way:
Then you would click “Cancel” and then switch to the MDI Tab.
Here you would first enter the offsets (they came from probing). They were X=15, Y=25 and Z=-50. To enter them, you use the G92 command and you enter it into the command entry field of the MDI Tab, but with their sign changed (i.e. the Additive Inverse):
G92 X-15 Y-25 Z50
Then you click on the “Set Axis button” of each axis one after the other, and set each axis position to the value you saved with your photo or screenshot:
This way, without the axes having moved, you restored the settings from before the power outage:
If you are sure the axes did not move, you could run your program again. It would first return to its program starting point and mill air for the parts that had already been milled away, and mill the rest normally.
Unfortunately I can’t test it because I have only the Onefinity Controller here but with no machine attached to it. You can report back if it worked, e.g. by simulating a power outage by simply pausing and shutting down (shutdown, not reboot), then powering on again and proceeding as described above.
The other case would be where the power outage occurred while the machine was moving, and in this case you will have to first home the machine, then we would restore the offsets only, before re-running the program. It’s late so I can’t describe this case today.
And what we could do another day, is make no photo but extract the offset values from the log file 
Got to go, it’s 21:34 in Central Europe, good night!
PS:
The Onefinity Controller does not support starting on a specified line. This only would work by stripping parts of the file. But this is not necessary if initial milling air does not disturb you. Removing the first portion of a G-code file is not trivial (you got to make sure the modal commands at the beginning of the file are preserved, and later modal commands too) and you also had to know the line where to resume.