Unwieldy? Or unwieldly?
This is a tough one—because neither the correct word nor its root are all that common these days. We generally have an inkling of the meaning when we use it, believing (correctly) that it refers to clumsiness or awkwardness. But which is the correct word?
Let’s start with the root: wield.
According to my OED, the pertinent definition here is “Direct the movement or action of, control, (a bodily member, a faculty, etc.).” Another is “use or handle with skill and effectiveness; ply (a weapon or tool, now always one held in the hand).” Both come from the Old English wealden. Then there’s this etymology entry, which mentions “moving gracefully” as a meaning dating back to the 1520s.
Now that we know what the root means, let’s apply some basic rules of English word construction. If wield means “with skill and effectiveness,” adding the suffix -ly would create an adverb:
Swinging the battle axe in a downward arc, Steven wieldly cleft Aaron’s skull.
Not very pretty, though, is it? Partly because it’s somewhat redundant. If wield already means “with skill and effectiveness,” we don’t really need an adverb form. But if we did, it would probably be wieldily, wouldn’t it? It stands to reason, then, that unwieldly isn’t an option.
Wieldy, on the other hand, starts to make more sense—especially when the prefix -un is added. As an adjective, unwieldy would therefore mean “not (un) under control (wield),” or “without effectiveness,” or “ungraceful.”
English doesn’t always make sense. But when it does, it’s a beautiful thing.