The symptom of a faulty solenoid is lots of pressing of the starter button with plenty of clicking at the solenoid with the starter eventually engaged and the engine firing up!
Before changing the solenoid, check the starter switch which is easily dismantled. Clean the contacts with fine wet & dry paper and gave them a spray of contact cleaner. If that cures the problem then you are done!
Less likely is sticking brushes in the starter motor. A simple but not conclusive test is to give the starter a motor a sharp tap if its not engaging. If that makes it engage then sticking brushes are probably the culprit. See here
I was still only getting a click when pressing the starter button after cleaning it so the culprit seemed to be the solenoid.
Replacing the solenoid is a nice clean and simple job thank goodness.
Tools needed:- no special tools are needed but 8mm and 10mm sockets will make the job easier.
Parts/supplies: New solenoid - seehere- which looks like this:
Remove the side panel and battery. Then remove the three 10mm bolts holding in the battery tray and take out the tray.
To remove the old solenoid, undo the 8mm bolt that holds it to the shock reservoir bracket and the 10mm nut that secures the black starter lead.
The old solenoid can then be removed after disconnecting the feed from the starter switch (single bullet connector) and the live feed going back to the loom (two-way block connector) as shown in the photo below.
Disconnect the thick red positive battery lead, pull the old solenoid and mounting rubber off the metal holder and fit the new one. Then re-attach the red lead.
Reconnect the two connectors and the solenoid is ready to bolt back in.
Once back in place, replace the 8mm bolt and fasten the black starter motor lead onto the new solenoid with the 10mm nut supplied.
Fit the insulator back over the terminal, and then refit the battery box, battery and side panel.
The new solenoid did the trick for me and the TTR starts on first press of the button every time now