Hi, I found some burnt wires in the engine bay harness of my Mk1 Esky. There are 3 I need to replace and I need to know what amp wire for each. The coil wire from the firewall, the wire from the coil to the starter solenoid and the horn wire.

If you have samples, take them into autostore and compare cross sectional area of the copper. Beware the cheap stuff that uses a thick insulation but has minimal copper

Good point with the cheap wire, been caught before. These wires have been replaced before with whatever they had laying around. Cooked the wires and tape.

All high amp circuits but do as others have suggested and get an auto sparky to supply equivalent stuff.  Being in the engine bay you need to ensure that the plastic coating has the required temperature rating.  Some of the Narva stuff available in small reels might be a bit iffy.

All the fuses in my MK1 are rated at 8A so you could just go for 13AWG or 2.5mm2 which will do the job. 

The horn wire has been cut short so I will run it near the battery to a relay to take the load off the switch. The black coil wire has been replaced almost from the firewall but the replacement wire cooked so I will replace it, the original black wire seems a bit light for the job. Can a relay be used here as well and take the load off what remains of the original?

If all circuits are protected by 8A fuses I would just trace all the dodgy cabling back to where it meets the original size, then use the blue (2.5mm2) insulated inline crimps to join 13AWG cable. Then take the new cables where they need to go

crimps will only cause you headaches later on, do it properly & solder it

Where I can I have shortened the wire and will take it to a relay and new wire from there. The others will be a solder joint with heat shrink. I wont be re-tapping it but using a split conduit.

shannon
crimps will only cause you headaches later on, do it properly & solder it

What, 40 years later lol?
Chuck a bit of heat shrink over it if you're that concerned.