ASPxPopupMenu is a server-side control, so its contents can only be changed in the server code. Thus, it is not possible to add / remove elements using only client code. However, there is a workaround. You can add all the possible elements to the menu during development and control their visibility in the client code. For example, this can be done in the PopUp event handler on the client side of the menu:
menu.GetItemByName("someItemName").SetVisible(false);
Here you will find client-side documentation for ASPxPopupMenu:
ASPxClientPopupMenu Users