In ancient Greek it would be εκάθαρας. καθαίρω = cleanse comes from καθαρός and has nothing to do with καθ+αίρω.I believe it simply means "(εσύ, probably Hercules) καθάρισες* την Ήλιδα από τις κακές της οσμές", in the Subject-Verb-Object-etc. English word order, and you just got it in a complicated way.
*καθήρες (<καθαίρω=κάνω κάθαρση) in ancient Greek.
You are completely wright.In ancient Greek it would be εκάθαρας.