If you swipe right on an article that's in the "Read Later" file, you'll see that it brings up a red Archive button. Press that button.
Then, go to the "Archive" line in the left column. That same article will be there. Swipe to the right again and this time the red button will read Delete. Press it and the article is gone.
Hope that helps.
Marilyn
P.S. There's also an option, under Instapaper's settings, to make it so Delete is the first option versus the second (e.g. removes the Archive function).