ok I am going to enter this debate both feet first! corrosion around the relay is the problem! This causes voltage drops which will generate heat! Very carefully clean up around the relay bringing the contacts to a shiny state! rust, gunk and corrosion are you enemies as is water ingress. I use a stranded escort cosworth fuel pump in my fiesta and it draws 5 Amps, nothing excessive. so the wiring should cope, unless it has been fiddled with and or corrosion has got in to it. -> immobilizers badly installed can stuff you right up as I have seen them installed on the power side of the relay!

Your relay is a coil of varnish coated wire that has had 20 years to cook its self and if water has got into the contacts on the power side it can get very hot too, so an OEM or better brand new one is a must! ( remember ford have been known to up-rate some of their relays and not shout very loudly about this so early models will not carry the load latter ones do) (see fan relay mk3 fiesta).

check the resistance and of the wires should be very small and voltage at the pump with it running!

PM me and I will get back to you if you have any problems.