I've mistakingly bought a pile of rubbish instead of a road bike (was going to be my first one), however, I'm determined to make the most of it and treat it like a project for me to work on. Currently it's right hand shifter is a 9 speed one (campagnolo), a broken rear derailleur (shimano) and an 8 speed cassette... with only 7 of the cog things actually present... (key words and knowledge in this area are pretty poor for me so I apologise.) I'm going to replace the derailleur, does this need to be a specific one? And then the main question... Is it possible for me to swap to a 9 speed cassette from the 8 speed? Would this make the shifters work effectively, to save the cost of replacing them?
Everything here depends on what rear hub you have, specifically what type of freehub body it has (the part that the cassette mounts on). Given that it seems to be a frankenbike, it could be a lot of things.
Also, given that it's a frankenbike, it would be wise to first double check that your shifter that says 9-speed is still set up for 9-speed, since Campy can be upgraded. So check that it has 9 positions (8 clicks).
The first thing to do is identify what type of rear freehub your hub has. Campy and Shimano each have different freehub spline patterns, and each have at least a couple different generations of freehub lengths dictating how many cogs will fit, and, in Campy's case, derivations of the spline pattern.
If you have any Shimano-compatible freehub, you can't set up the bike with any kind of normal drivetrain and would need to resort to some kind of compatibility hack. There are actually a couple pretty reasonable choices there, unless you got really unlucky and it's a 7-speed Shimano freehub. Namely you could upgrade the shifter to 10-speed and buy a 10-speed Campy rear derailer and 10-speed conversion cassette, or you could upgrade the shifter to 10-speed and then use the "Hubbub mod" with a Shimano RD and 9-speed cassette, which is a much cheaper plan.
If it's a Campy-compatible hub, you need to figure out what generation of freehub body it has. Campy 9 speed cassettes don't fit on their 8-speed freehub bodies. Campy used to sell upgrade 9-speed freehub bodies for their 8-speed hubs, but I believe they haven't been made for a while and may be fairly unobtainable now. If it's a 9 or 10/11 speed freehub body, you just need a Campy 9-speed cassette and rear derailer.
Note that buying a bunch of Campy parts to make your Campy shifter work may be far from the most economical plan, especially depending on the condition of the Campy-compatible wheel if that's what you have.