if you look at the traces, you can see the charger plus wrapping around the end, with two of the pins of the cermet soldered into the trace. the other leg of the cermet does a dogleg around the led and the led resistor R40. that dogleg is attached to the second resistor R39 if i am correct.
the other end of R39 is connected to a trace that goes back to the 14 pin IC which is the TL494. that resistor R39 is 1600 ohms as i recall. your cermet trimpot when adjusted to the max got to 14V. so total of 1600 + 2,000 is 3600. but when it was 1,600 only (when you adjusted it down to 10V) so 4V out of 2k ohms. you wanna get to 24V right? or 22V? or 20?
if you have some small resistors around 4-5k ohms, you can try sticking that in the place of R39. maybe here is a good time to determine your soldering skill too.
if you have little jumper wires with tiny alligator clips on the end, you can unsolder one end of that 1600 ohm resistor brown green red R39 as i recall.
then solder the 4k ohm (or close 4k-5k ohm) resistor leg in the hole you pulled the old leg from so you have the two resistors standing up. then connect the jumper from the up leg of the 4k to the leg that goes into the trace on the 1600. plug it in and test the output voltage, if not close, then we can select another resistor for the replacement. and you can jumper across the top from one to the other to put them in series and test. or if you have a 6k ohm resistor you can try that size to start, somewhere in that range since we have the trimpot.
when you are working with the resistor sticking up like that, it cannot touch other stuff either, but you can test the voltage without reassembling the entire charger. just put it on a surface that is insulated, the transistors and the diode don't need to be heatsinked since they don't supply much current when you are just testing the voltage. make sense?