melt the solder at each junction and test again. the gunge is probably flux from resin/flux cored solder used previoulsy it will look like brown wax it chemically burnishes the joint as it melts and gets all over if you use big fat powder cored solder
if all good job done
or do it properly
desolder use solder sucker or desoldering braid
pull through
be careful not to pull the track off the board
use plumbers flux or soldering flux powder on the copper and the pins, heat it and it flows and cleans. (or just give it all a clean up and use flux cored solder)
give the mess a mop up but don't touch the track with greasy finger once it has been fluxed it kinda acid etches the surface for a good solder flow
tin both with solder
assemble
light touch at the junction with the iron to flow the previously applied solder down the pin and into the hole and across the track
make sure you don't link anything with a solder bridge that should not be linked
test it again.
the best joint is mechanical joint pin pressing on copper that is glued in place with solder.
when the high end hi fi guys build an amp they make connections by twisting their components onto tag boards
the amp would work without the solder. its put on as a permenant way to make sure the physical connection stays put. that way the conductivity/resistance of solder doesn't become part of the amp circuit. only the component's rated conductivity is included
if that method is good enough for them its very very good, possibly to the point of overkill here.
dave