that is the starter solenoid making that noise.... It's isn't getting enough voltage. It could be a bad battery, Low battery, Loose ground connection at the battery or frame, or a lose positive connection at the terminal or solenoid.
check all of those for tightness and cleanliness. or your valves need adjusted on the engine...
Do you have a volt meter?
I'm going to steal a trick that i first heard from Bert, if you have jumper cables, hook the positive to the + on the battery, and the ground to the - on the battery, other end of the - cable to a good clean ground on the mower... drain plug, bolt, etc. and then touch the + lead to the starter terminal, see if it will crank. expect some sparks. your starter should look similar to this one i posted a picture of, you'll see where the positive wire should touch.
also post all identification numbers, there will be a mower model number under the seat, should start with 917.xxxxxxx or sometimes another 3 digits. and post the engine Make and we'll tell you where to find your Engine model #
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