Copy functionality designed in such a way like that field values that are populated from the original record that was used to copy
One solution you can try that create a Process Builder flow to clone the ticket; where you can have control over which fields coped or not
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You can possibly add a custom workflow on criteria IsNew() --> i.e. when the record is New
Set an Action to set the the custom field value = ""
The only catch is you wont be able to directly differentiate between a new record getting created from Clone or directly from the form itself.