The "On" button doesn't turn the printer on suddenly....

I had a similar problem with an HP OfficeJet All in one that drove me nuts.It would loose power,then come on, then loose power again.What I figured out and you can investigate is that although the power adapter (mini ups) is on green light the connection is not making contact.If you play with the power plug on the printer, jiggle it, pull it out,push it it and you find the printer starts,trys to go into setup thats your problem.I tried using tape to hold it in (worked for a while),tied the cord wire up relieving stress on the connection,(worked for a while till cat thought it was a toy).My best solution was to add tiny tinfoil arround pins or holes,(lasted months).I since upgraded to an all in one wireless.I don't like the adapters that HP uses on its printers.They should by now make a better connection detail.Sorry for the long response I still get angry as I'm sure you are but if this is your issue you can fix it.Good Luck.

If the printer is plugged into a surge protector, you may want to unplug it from that surge protector and plug it straight into the wall.

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The other thing that comes to mind is that some printers have an internal battery that looks like a flat disc about the size of a nickel. If this battery goes bad, the printer may not turn on. I could not find any material online that suggests a battery exists in this model, but it's worth a look inside the printer anyway. It probably will be in an "out of the way" area under the hood with the cartridges and it shouldn't be too inaccessible.