It is an lm723 based power supply, maybe finding some schematics on google may help you
something like :
http://chemelec.com/Projects/LM723-PS/LM723-0.png
http://www.electroniq.net/power-supply/lm723-variable-power-supply.html
some basic implementation
http://www.rason.org/Projects/powsupply/powsupply.htm
Looks like that burned resistor services the Vcc pin on the 723, as well as something off-board (there appears to be a blue wire connected at the opposite edge of the board via a PCB trace from this resistor). My guess is that it's a low-value resistor. Unless you can make out its value (or have a schematic available) I would be tempted to replace it with something like 10 ohms, and also replace the 723. Then check the 2N3055 for damage and replace it if necessary. Also trace that blue wire and see where it goes. After repair, power up slowly with a variac or something similar if you have one and check to see if the 723 or the resistor (or anything else for that matter!) is getting hot. It's worth a shot and probably won't make you any worse off that you already are!
The fact the output voltage dropped from 21V to 5V when you removed the burnt resistor means it cannot have gone open-circuit. There's a fair chance its somewhere near its original value. Its resistance may have gone up a bit but measuring it may give you some idea of a possible replacement.
I strongly suspect that the 2N3055 pass transistor has failed. While its supposedly good for about 5A with 23V across it, and I *ASS*U*ME* the PSU has some sort of fixed current limiting that kicks in at around 2A, that 5A rating form the S.O.A. graph in the datasheet is only applicable with the heatsink at 25 deg C. Put a sustained short on the output and the transistor heats up the heatsink till it can no longer take the surge when you short it.
It would also be worth checking that relay - I *think* its a tap changer to select a lower input voltage to the bridge rectifier at low output voltages, and if its contacts have welded in the higher voltage position, that would result in pass transistor failure.
If you need to reverse-engineer it to get a schematic to help us work out a likely value for that resistor, you'll need to take well lit, in-focus photos of both sides of the board, square on, not skewed, and preferably against a white background, so we can help you overlay them to see the components on top of the track layout. You'll also need to identify what each off-board wire goes to, and make a note of all component part numbers and/or values that can't easily be seen from the photos
Reverse engineering is hard work so lets try the simple stuff first.
Looking at the underside of the board again: is it just me or is there a bad solder joint on that relay?
I presume you're referring to the solder joint next to the screw on the right? If so, there's nothing connected there for whatever reason
A.K.A. Skytronic 650-673, this unit appears to be an over spec'd clone of a Velleman PS1502.
As far as I can tell, it corresponds to the schematic given on the last page of the Velleman PS1502A user manual. There are a few minor component value differences around the voltage feedback/control - as you might expect when the output is specified to 20V. R13 appears to be extra and is in parallel with R8 (! uh).
https://www.velleman.eu/support/downloads/?code=PS1502A
Is it my imagination, or is that screw on the left on the backside bridging two solder pads?
The 3055 can be tested for shorts with a basic multimeter set on the lowest resistance range. If you have a meter with the diode check function you can probe with positive lead to base...and with negative lead to collector or emitter you should get about 0.6v. Any reading in any other position would indicate leakage or a short. The 723 would be hard to properly test. Since they're cheap I would just replace it as a matter of course.
Nusa has a point here, though, about that screw....
A.K.A. Skytronic 650-673, this unit appears to be an over spec'd clone of a Velleman PS1502.
As far as I can tell, it corresponds to the schematic given on the last page of the Velleman PS1502A user manual. There are a few minor component value differences around the voltage feedback/control - as you might expect when the output is specified to 20V. R13 appears to be extra and is in parallel with R8 (! uh).
https://www.velleman.eu/support/downloads/?code=PS1502A
AARGH!!! Its a PNP* pass transistor design without effective current limiting - LM723 pins 2 and 3 aren't even hooked up, and there doesn't appear to be any other active current limiting circuit!
* The datasheet the O.P. linked includes the PNP MJE2955 as well as the titular NPN 2N3055, and board layout doesn't match any conceivable NPN pass transistor LM723 variant unless its got a PNP driver transistor off-board soldered directly to the pass transistor pins as a Sziklai pair.