One thing as an Admin you have to be sure of is that your sub-users have all of the correct permissions. This article will take you through that process.
You will first want to go to your Settings and find the Users & Permissions tab.
Then you will want to click on the "Click her to configure security permissions" link to pull up your sub-user's permissions.
You will want to be sure that your sub-user has either All Sections- Delete or Contact- Delete based on the specific permissions you want to give your sub-user.
Once you have updated those permissions, be sure to hit the blue 'Save Permissions' button.