From "Japanese for Busy People Level II" p 46, there is the following question:
Fill in the blank with the appropriate particle:
この くつを （ ）みても いいですか。（はきます）
In other words, modify はきます so that the sentence makes sense when the modified word fills the blank.
This book is slowly introducing kanji, so you can see there are none here.
I know: このくつ mean these shoes. はきます mean to put on. いいですか。 is asking permission whether it is okay.
If the みても were not there, I think the sentence would make sense if te-form (はいて) filled the blank. But the みても is really confusing me. I really don't know what it means! I suppose it's just te-form みます (to look). But, I have no idea what it's doing in this sentence. Also, the particle も doesn't make sense to me.
I did learn something recently, which may or may not be related, even though it's not exactly helping me here... That is, using a te-form verb followed by みます, means "to try to do that verb." But in this sentence, it seems opposite. In other words, if I'm not mistaken, one could say, "～をはいてみますいいですか。" Which would mean, "is it okay if I try to put these on?" I could easily be mistaken, especially with particles.
Sorry to ramble, but I'm just trying to explain my current level of understanding.
Thank you for reading.