It looks like it had water soluble flux residue remaining on the PCB and then got wet, causing metal salt conductive bridges.
You may need something more aggressive to clean that off. I usually use (because I happen to have some) brasso metal polish and a toothbrush. It's the mild abrasive particles in it that help more than the oxalic acid. After scrubbing with the polish, then I'd scrub with hot detergent solution and a final water rinse. If it had fine pitch SMD chips I might do a final rinse with alcohol but for that, I wouldn't bother.
If after scrubbing all that corrosion off, there is bare copper left behind, you might want to paint it to seal off from oxygen, whether it be with a lacquer, conformal coating, nail polish, or anything really is better than nothing, except some coatings could interfere with any later repair efforts and anything opaque, also interfere with ease in tracing the circuit tracks.
However, I wonder if this is coincidental, that the real issue is one speaker connects to the other with a 3.5mm or similar jack and the spring tension contacts in the socket have gotten loose or tarnished a bit so the manipulation of the cord, even if the power cord instead of the link to the other speaker, is moving the plug in the socket just enough to restore the connection.
If that's what's happening, you could of course source a replacement socket with same form factor, or do without the connector and solder the wires directly on, or since there is so much empty available space on a speaker housing, just put a hole in the housing for a panel mount jack, air-wired to the PCB contacts.
Otherwise I don't see how wiggling the cord would be doing anything about the corrosion on the PCB. If the power input socket were intermittent that should affect both channels.
