RE: How can I correct this payment so that it shows as payroll and the proper taxes are withheld?
You can do this by creating a gross-up paycheck:
1 - If the check was recorded in QB as a standard check, delete or void it.
2 - Create a new paycheck for the employee. As you do, on the first slide of the create payroll process, click the employee name to open the Preview Paycheck window
3 - Then click the Enter net/Calculate gross checkbox and then enter enter the desired Check Amount as pictured below.
QuickBooks will respond by grossing up the paycheck from the net amount you enter, as pictured:

Note that you have to use a Salary or Bonus type earnings item in the earnings table to do this.