Post #13 is key here. He verified in that post he's seeing a 6V drop in the positive chain when trying to start. As others have mentioned, you need to take your meter and ground the - lead and work your way back from the solenoid to the battery and measure the voltage on each side of every device WHILE THE KEY IS IN THE START POSITION, in other words, under load. The voltage across all safety switches, PTO switch, ignition switch etc that are in series between the battery and the solenoid need to be measured. In a perfect world, only one of those devices is causing the 6 volt drop and that's what needs to be replaced. However don't be surprised if you see a couple of volts being dropped across multiple switches. Then you have multiple contributors to the problem. To be clear, the readings are meaningless without the load of the solenoid coil with the key in the start position. I promise you Ohm's law is still in effect here!
As Star Tech mentioned, you may find it easier to put in a helper relay. I did. I have a JD170 tractor(with Kaw motor) that was prone to this clicking problem and I had no desire to replace all the intermediate safety switches, ignition switch etc. Hasn't clicked once since. [google AM107421 for more info] This is the fix JD came up with as a field retrofit. In a nutshell you are using the trigger wire to power an external relay you locate near the starter assembly and it switches local battery power to the solenoid. The external relay is a much lighter load as it has a coil resistance of say 80 ohms drawing only 150mA as opposed to the solenoid which requires a number of amps which is over 10X more current. There is a much lower voltage drop through whatever extra resistance is in your trigger wire 'chain' (i.e. dirty/corroded switches/wires/connections) when 150mA or so is flowing vs the solenoid amperage.