I tried with another power adapter (the U64 one, has exactly the same plug, 12V an 2A).
The machine ran stable for about 7 minutes, after that, going on with resetting intermittently after 10-200 seconds.
There was a problem with reset glitching, which we believed to have universally fixed. However, it looks like yours is very unlucky.
There are some improvements to this in the latest core, which we should find a way to get on your machine for you.
The problem is a temperature related one (I can go into the details if you like, but not required just now. It's not a hardware failure, but rather a logic edge case when the main and MAX10 FPGAs clocks drift too far apart). This is also likely why it was stable for several minutes on first turn on, as the MAX10 FPGA would have been cooler than after running for a while.
So, assuming you don't have a TE0790 JTAG adapter handy, turn the machine off, and have it somewhere cool (but not in the fridge, or somewhere like that, which would result in condensation), and you should get longer on first run, hopefully, long enough for you to do the core update, which should be more stable. Alternatively, a can of freeze spray and giving a quick squirt to the MAX10 should be broadly equivalent.
If the problem persists, we can look at further solutions. One way or another, we will get you up and running stably. Email me direct at firstname.lastname@example.org if you wish, as I am not keeping up with all the traffic here at the moment.