"Could you clean up the room?"
Grammatically, the 把 construction requires a compliment after the verb (at least, that's what I was taught.) So you can't end the sentence with 打扫. You need to put some kind of complement after it. (I'm pretty sure 吗 does not count as a complement.) A complement is anything that gets tacked on directly after the verb in Chinese that further explains, specifies, or otherwise modifies it.
打扫 is the verb "to clean" 干净 is the adjective "clean" (describes a thing that is clean.)
In 打扫干净, 干净 is a resultative complement. It describes the result of 打扫. 打扫干净 means "to clean until the thing is entirely clean," as opposed to cleaning for 5 minutes then stopping to watch a movie.
I believe that Duo translates 打扫干净 to "clean up" (instead of just clean) because "clean up" has a similar sense of completing the cleaning.
Should one use the structure with 把 or is it proper to have verb —> modifier —> object: 你可以打扫干净房间吗？
Using the 把 structure in this case is certainly correct. I'm honestly not sure whether your sentence is also correct or not. However, the 把 sentence sounds more natural to me.
This is just how I personally think about it, but I think of the 把 structure as a solution to this situation where you have two things that ought to go right after the verb. Using 把 gives you another place to put the object. It sort of unclutters the end of the sentence.
My memory is vague but the bă sentence construction usually has the bă followed by the noun then the verb and then followed by the desired result of the verb's action. Not always but it's tidy grammar. Something like that. Can somebody clarify/confirm/otherwise that?
You are exactly correct: in formal Chinese 把 specifies the object of the verb. It's typically not used in spoken Chinese, where this sentence probably would have been translated as 请给房间打扫一下.
I think 扫 is just a non-specific verb... like wipe or sweep. Might need the 干净 to make it about cleaning? Not sure.
It should be acceptable. Although I think that would be closer to "clean" instead of "clean up".
Judging from their use of non native English, I don't think they know the difference between clean and clean up. Albeit small but there is one.
That sentence is grammatically fine. I think Duo just wants you to use the 把 construction. Also I agree that's closer to "clean" than "clean up" ("Clean up" has a sense of completion to it, like you've finished the job, which is what 打扫干净 means.)
Can someone help me understand why 你可以打扫干净房间吗？would not be correct?
Besides the accepted answer possibly placing more emphasis on "cleaning up" by moving "打扫干净" to the end of the sentence I believe 你可以打扫干净房间吗 should be an acceptable alternative.