Mower will not start. Makes a loud fast clicking noise when the key is turned.

pull the plug out and see if it will turn over it could have overheated causing it to lock up if it is look at buying short block are new motor but befor doing this how old is it is it worth buying a new motor

check for vapor lock

well if it turns over with solinoind jumped out and not with the key id say yes your on the right track as it's something in the safe to start circuit could those wires you found posibly be near the seat maybe if so I'll bet they fit in a seat switch that may have got unhooked for something

If you crossed the two battery terminals on the solenoid and the engine did not turn over, I would check the battery terminals first to see if they are tight. Then check the terminals at the solenoid to see if they are tight. If all of this is tight and you are sure you have 12 volts at the solenoid, then your starter is bad.

check your wiers that go to you solinoid but first try to jump your mower usind jumper cables put the red cable on the bottom bolt of your solinoid and find a good ground for the black cable then let you car warm up then trun the key on your mower to start and it will start

Something broken in starter. Remove and check or have shop check it.

no,,just turn the key to start her up!,,,if the engine is free to turn i gess it has a stuck starter relay or a jamed starter, or a dirty power/earth lead or joint some were
but check that this battery you charged up is any good first if the battery has been left standing in a discharged state it wont be any good any more,,try a set of "jump leads" from a good car battery!

The model numbers you show are to a Tecumseh engine. If this is what you are searching for, you can go to www.outdoorpowerdistributors.com.
There you can get a pictorial with part numbers.

definately connection some where i would use a volt meter and make sure where power is and make sure it is getting too where you need it, if its getting too the soliniod or not then too the starter.

Assuming the pinion gears and wheel gears match, (sometimes dealers make mistakes too.) Check the bearings behind the wheel adjusters and the retainer-bushing Hoc. If they are in good shape lubricate the pinion gears and wheel gears w/ hi temp lithium grease.