How do I start a clam to dispute a charge made to my Apple Cash account

Transaction number will not help in this forum as we have no access to your account. Unsure of the reason for the dispute since you said it was made with your permission or who it was sent to. When sending Apple Cash to another person, in some cases the only recourse is to ask them to send the money back if they have already accepted it. When using Apple Cash, it is best to only send money to people you know, since you are not given the same protections as with a Credit Card. Speaking with Apple directly will be the only way to find out your options:

Contact Apple for support and service - Apple Support

Also for more information on sending money with Apple Cash and steps to take to cancel or dispute a transaction:

Send and receive money with Apple Cash - Apple Support

Apple Cash is a service that’s administered by Green Dot Bank. Contact an Apple Cash Specialist at Green Dot Bank by calling (877)-233-8552 or by calling 1-800-MY-APPLE

https://applecash.greendot.com/termsconditions/