I was having this same issue. My mower was running but needed a new battery (tested with a meter and was reading low...the mower sat up for a long time without being ran) so I had to jump it off a few times. In the process of cranking one time, it started making the clicking sound you described. I have some experience working on engines, vehicles and mowers, so I started checking the obvious things...battery, solenoid, safety switches, electical connections, etc. Eveything checked out fine. I figured it had to be the starter so I took off the upper engine shroud to get a better look and saw that the starter spline had got lodged on the flywheel. All that was needed to do now was to unlodge the starter spline allowing it to drop down. I left the shroud off and turned the key and there she blew!!
Let us know if/when you get it running and what the issue was!