HDS2135 starter sometimes clicks, sometimes starts

I have a Cub Cadet with a Kohler CH18 engine. Sometimes I can hear the solenoid clicking but the engine wont turn over. If I hit the key 4 or 5 times it will come to life, starts and runs fine. I have a fresh battery, shows 12.8V at the B+ post on the solenoid, when I hit the key I am showing around 7V at the solenoid (spade post). I have pulled all positive and negative connections and cleaned them up. Pulled the starter and bench tested it, so far it has never failed, both by direct contact with starter and running a jumper wire from the solenoid. Could I just have a dead spot in the starter? I don't think it is a key switch as the voltage is coming through to the solenoid. I think it is 1 of 3 things, starter motor, solenoid, or switch. Don't want to dump a ton of money into chasing the offending part as I am getting ready to sell it.

Slight update that may help. I tested the small solenoid lead when the starter was off. With the key turned to start position I am reading 12.1V. I reinstalled the starter hit the key and it just clicked. I took a reading at the same lead with it plugged in to the starter and it was 6.1V, so I pulled off the wire and test it not connected to the solenoid again and it was showing a full 12.1V, hooked it up again, solenoid clicked and only showed a reading of 6V. Thinking I may have a solenoid problem.

I owned a 2135 for many years and the switches and wiring were always troublesome. I initially replaced the starter switch and the PTO switch with OEM parts, but ultimately, the only way to keep it operating was to rewire the entire machine with fewer but commercial-duty switches. It's up to the individual owner to decide if he is comfortable without the brake, seat, reverse and PTO "safety switches". I was and from that time on, the machine never failed to start and operate. FWIW, there are only about four wires necessary to start, run and cut.

jack vines

If you are getting a 50% voltage drop with no load other than the starter solenoid then either you have very dirty connections between the battery & the mower or your battery is deceased.

Get the battery load tested.
Cheap & nasty method is to hook a high draw thing line a headlamp globe across the battery and measure the before, during & after voltages.
Anything more than 0,5 V and the battery is toast.