I'm not familiar with the HM604, but I have worked on HM203-5, 203-6 and 203-7, HM605 and HM1005.
You need to download the service manual for the HM604. There are a couple of versions, so you will need to download the one for the version of HM604 you have. With Hameg manuals the German version usually has circuit diagrams and PCB layouts, other language versions may not.
This is one site where HM604 manuals are available, but there may be others, which may offer better copies.
https://sky-messtechnik.de/reparaturdienst-service/downloadsI'm not completely sure what exactly the wire with the detached pin is. It looks as if it's a pin which was soldered to the PCB and which had a wire soldered to it. You would need the service manual to know what it did and that you were putting it back in the right place. it looks as if it was soldered to a PCB track which has lifted from the PCB but that's odd, in that such pins are usually through hole. Repairing detached or burned copper tracks can be done in various ways, but usually a thin insulated wire replacing the track will do, but it's always messy.
Usually nuts on these things need to be comfortably tight, but no more. If they are loose they are not holding anything together and could work completely loose, with a chance of causing a short circuit.
Again, I'm not certain what the last photo represents, so this is largely guesswork. I suspect it's the final Y amplifier. The white discs with nuts are probably heat conductive, insulating spacers for transistors to be attached to heat sinks. There may have been a cover which is missing, which would have acted as an electrical screen and also as a heatsink for the transistors. The nuts would have secured the spacers and transistors to the cover. The cover would have access holes so that the trimpots and trimcaps could be adjusted.