Another possibility to consider.
There are a number of safety switches between the start terminal on the key switch and the terminal on the starter solenoid.
As these switches age, they develop a little resistance to current flow. Several switches with this condition can case enough of a voltage drop at the starter solenoid "kick it on terminal" that the solenoid does not engage forcefully and does not make a good enough connection to turn the starter over.
Two ways to investigate this.
Remove small wire from the solenoid when it wont start and apply power to the terminal directly from the battery using say a 12 gauge wire. Remember you are by passing all safety features when doing this. Be extra careful. Have someone on the machine so it does not start and run you over.
Buy a cheap $15 multi meter. Measure the voltage on the "kick it on terminal" on the starter when you have the key turned to start and it just clicks.
If you are unsure about the terminals on the starter, post a picture and we can help you understand it further.
Garden tractors are notorious for developing this condition over time if the owner lets the machine sit outside in the rain. The safety switches on these machines are very cheap and have no water or mud protection to keep their contacts clean. I don't know about the quality of Mule components.
The multimeter will also allow you to verify the charging system. You can also check for voltage drop in your battery cables without having to clean each end of both just in case.
This link to DENSO starter trouble shooting tips shows you how to approach the problem in a methodical way. The illustrations near the end showing how to check for poor cables or connections are invaluable if you have even a cheap multimeter
http://densoautoparts.com/sites/default/files/tech-tips/DENSO Starter troubleshooting tips.pdf
Dave
M7040