No, permit requests cannot be "deleted" as such. However if there is a permit request which needs to be cancelled or is not needed, it can be removed from the main Dashboard list by changing the request's status to "Permit Not Required".
To do this, open the relevant request's Job page. Assign it to yourself (if it hasn't been already). From the status dropdown bar at the top of the screen, change the status to Permit Not Required:
A popup window will appear asking for a reason for changing to this status - type in the reason, then click Save:
This permit request will now no longer show on the Dashboard list.