I have the same identical problem with my '06 TRX350 Rancher. Replaced the battery and starter relay. Still clicks. Pull/rope start works fine. The starter is next. I guess I could have saved some time if I read deeper into the issue or followed the checklist procedure in the service manual.My '04 350 Rancher won't start with the button all the time. If I pull the starter cord through, eventually the starter will run and the engine starts. I keep the battery on a battery tender and even off the tender for a couple days, I'm still reading over 12 volts, usually in the 13 range. I changed the solenoid and still have the same symptoms. I am thinking, as another poster said, that the starter is the failure point. All contacts are solid and clean.