Services Module Unable To Update User Fields
Way 1:

Yes, you need to send the current password to update the user email that is default Drupal user account configuration, and the default user update enpoint exposed by services programmtically submit the same user profile form, which triggers all the validation your form has. In order to send the current password your request load may look like

{
    "current_pass": "onso",
    "mail": "test@example.com"
}


Way 2: 

If you don't want to provide current password, you can consider using No Current Password module, but remember that it will remove the current password field from the form UI as well.

Way 3:

Expose custom endpoint to update user email. Make sure you have authentication on your endpoint.