Hi Shane,
My computer was an end of line sale item, (marketed under the 'Vulcan'  brand name)  in one of our UK stores - John Lewis. It is now about 11 years old but still doing all I want. (apart from this unresolved problem)   
I managed to get a screen shot of the motherboard spec, from Piriform Speccy program which shows an 'Unknown slot' designated as AGP.  It shows it as 'In Use' but I don't know what uses it.  I have attached the screen shot here.  
It doesn't mean anything to me but it may give you the information you need.
I really appreciate your advice.  I have been trying for a month or more to get to the bottom of this but no one else has got so far. Many thanks. 

Motherboard
   Manufacturer   MICRO-STAR INTERNATIONAL CO., LTD
   Model   MS-6378(VT8361) (Slot A)
   Chipset Vendor   VIA
   Chipset Model   Apollo KLE133 (VT8361)
   Chipset Revision   00
   Southbridge Vendor   VIA
   Southbridge Model   VT82C686
   Southbridge Revision   40
      BIOS
         Brand   Award Software International, Inc.
         Version   6.00 PG
         Date   18/12/2001
      PCI Data
            Slot ISA
               Slot Type   ISA
               Slot Usage   Unknown
               Bus Width   16 bit
               Slot Designation   ISA
               Slot Number   0
            Slot PCI
               Slot Type   PCI
               Slot Usage   Available
               Bus Width   32 bit
               Slot Designation   PCI
               Slot Number   1
            Slot PCI
               Slot Type   PCI
               Slot Usage   In Use
               Bus Width   32 bit
               Slot Designation   PCI
               Slot Number   2
            Slot PCI
               Slot Type   PCI
               Slot Usage   In Use
               Bus Width   32 bit
               Slot Designation   PCI
               Slot Number   3
            Slot UNKNOWN
               Slot Type   UNKNOWN
               Slot Usage   In Use
               Bus Width   32 bit
               Slot Designation   AGP
               Slot Number   4