You can't trash "part" of a clip. If you have a long clip and want to only keep part of it, you'll have to select a Range around that part, export it as a Master File, delete the original, import that new clip you just made.
Just to add…if you want to work with a lighter footprint, there are a few other things you might consider (assuming you're shooting with a file-based camera).
Use a card reader for import and set ranges for your clips in the import window. Only the selected ranges will be imported.
Or copy the entire card to the computer. Then import, but don't copy the media to the event. You'll have aliases pointing to the media in the folder (or img) that the card was copied to. That won't affect your editing and you will not have taken up additional space on your drive.