if the car is standard and doesn't have
1) electric fuel pump
2) electric fans on the radiator
3) super bass 300watt amp and a sub woofer
4) halogen headlamps
5) kettle for making a cup of tea
6) electric fire for those cold winter days
you get the picture basically stuff that would use way more current than standard
AND you didn't replace the orginal alternator with a 150amp sucker off a modern fuel injected vehicle
and you battery isn't way over spec and constantly half flat.
The wires got hot probably due to resistance of that connector where it goes onto the gauge
in which case
repair wire or its insulation first
clean the connectors, and make sure they go on nice and tight. if they bolt on with a nut use 2 spikey washers and smother the connection with grease
Then i'd go a check my headlamnp switch
and my igntion switch for similar hotness
the issue is really....
They never expected your car to last more than 5 years..
The connectors are exposed to the atmosphere
The connectors are made of a brass type alloy that includes a high % of copper potentially some tin and other metals to make a springly gold coloured alloy for the connectors
these metals when exposed to oxygen and humidity in the air corrode and gets a type of tarnish or rust on them called verdigris
white/green/black dusty tarnish
it doesn't conduct electricity unless its a bit damp meaning the contact patch between male and female parts of the connectors gets smaller and smaller as the greeness spreads into the junction, and a tiny contact patch carrying a high current gets mighty hot
its like you reduced the thickness of the cable at that 1 point down to a size that is trying to burn out under the load.
by cleaning up
1) ammter connector
2) ingnition switch connector
3) bulkhead connector
4) battery clamps
5) both ends of earth strap between body and engine and battery negative and body
you can make an improvment to performance of the car that you feel in the seat of your pants
small brass wire brush
small set of pliers for bending connectors to get better clamping
bag of assorted spikey washers
and a tub of vasaline
will make for a boring but successful weekend of car fixing..