There are many things that could be causing this. The first step is to figure out how that content is being rendered.
- Log in to your Magento admin backend
- Go to System->Configuration
- Scroll to the bottom and click Advanced->Developer
- In the dropdown menu on the top-left, select the store view that is loading on your homepage
- Expand the "Debug" section, uncheck "Use Website", and set "Profiler", "Template Path Hints", and "Add Block Names to Hints" to Yes
Now when you reload http://www.coveops.com (or hopefully your dev server instead), you should see a red highlight around each block displaying information on how that block was rendered.
One piece of information is the path to the template file being rendered, for example:
frontend/base/default/template/page/html/breadcrumbs.phtml
You can find that file starting in the app/design directory.
The other piece of information available there is the class name of the block that's calling the template file, for example:
Mage_Page_Block_Html_Breadcrumbs
These class names follow the format:
Namespace_Modulename_Block_Path_To_Blockname
So the above example would be found in:
app/code/core/Mage/Page/Block/Html/Breadcrumbs.php
Once you've identified the block and the template in your filesystem, you can read the code and look for reasons why the block wouldn't load when parameters are present in the URL. Most likely ?asdf is being taken as an argument to a method in the block, and it's causing that method to change its output, or output nothing.
Another place to look might be in your admin backend, under CMS->Pages->Manage Content, or CMS->Static Blocks.