How does the word " on " act in the sentence " i put my clothes on "?

If a word answers the question, how, when or where, it is generally an adverb. In this instance, I believe the word in question is a compound word, "put on", which means to get dressed. Therefore, I would say it is actually a verb, but could be construed as an adverb. Not to confuse you even more, but putting clothes ON and taking them OFF also could qualify as prepositional phrases, as on and off are prepositions. I can see why there is a dispute, because basically the phrase is more colloquial (used in everyday speech) than would be considered formal grammar. I guess I would go with the compound verb, though, rather than an adverb, but I could be wrong.