By complete coincidence, I recapped my HX-20 this week and ended up with the same problem. I used Mr. Birt's excellent Capacitor Map, cleaned the corrosion with a fiberglass pen and IPA, added flux and solder and then removed it with my Hakko FR301, and the new caps went in fairly smoothly.
Unfortunately, while the unit was working perfectly before, I'm now only getting 3-6V to the screen. There's a lot of variation for the first 1-60 seconds (depending how recently I turned it on), and then it just drops to a wobbly 3.5-3.8V and stays there. I hooked up VLD (Voltage going to LCD) to the scope so I could keep an eye on it, and it's not smooth at all. All continuity checked out OK, but just in case there was high resistance from corrosion or some other trace issue, I temporarily jumpered each trace with alligator clips on the LCD power circuit, while keeping an eye on VLD (which didn't improve). I spent way too long studying the schematic, and checked each of the pins of IC "2C" (it's getting 5.3V from the transistor Q8, and it's outputting a pulse, although I don't think it's outputting the correct voltage because by the time it gets to C2 (not to be confused with 2C) it's already down to 0.8V. I don't have a Chip Tester to test "2C" with, or a spare TC4049 - for reference, the modern equivalent is CD4049UBE. I did try reflowing the pins on "2C". I also changed diodes D5/D6/D7 with 1N4148 which didn't make a difference, and even ended up replacing C1 and C2 a second time just in case. Interestingly, removing C1 entirely doesn't make a difference at all. The area around it was corroded and I cleaned that up, but as I said earlier, continuity is fine for everything nearby, and I never saw any evidence of a failed trace.
In the end, I just bridged + of C6 to Pin 3 (5V) of CN7 (Expansion Connector). This turns on and off with the power switch, and it forces the screen to 5V, which is at least readable, and still adjustable with the contrast knob. It's not the ideal 7 Volts, but at least nothing appears to be overheating or shorting.