Veryu important is it a click or a klunk ?
There are two things that go click when you turn the ignition on.
1) the fuel solenoid, it is small so makes a light click
2) the starter solenoid which is big and makes a klunk.
And have you checked the oil level as asked back in post # 3 ?
Do you have a voltmeter ?
IF so what is the battery voltage ?
What engine dose your Dixon have in it ?
To shut down while mowing either the magneto shut down or the fuel solenoid shut down.
While there could have been mechanical problems usually they prevent you from rotating the engine by hand.
Things that cause the magneto to shut down while mowing are generally a short to ground of one of the safety switches , or a very flat battery if it has an ignition relay.
Things that cause the fuel solenoid to shut down are a loose / broken wire or a very low battery .