I cannot remove a part from the structure. I have the correct access rights and the owning item is in status work. However it says “Cannot remove object” 

This case may seem a bit confusing; often there is no references to a part, and given the conditions in the question nothing should prevent removing such parts.

However, in some cases a part may make a reference to another part, as a DefObj, or RefObj reference, (or in some cases even the node according to the part). And in order to remove the referenced part the referencing part must be removed first.

You can always identify such references using the References view. The view lists all kinds of references including the one described here.