I can't re-download a song with iTunes on my PC or iPhone

I had this exact same problem and I tried the same solution you did: deleted the file and could not redownload. It just said "Purchased" and there was no option to redownload. It turns out that I must have selected to "hide" the purchase. I fixed this by going to my account settings in itunes and clicking "view hidden purchases". Low and behold, my missing file was there. I clicked the "unhide" button and was able to download it again from the itunes store.

I found the solution!

You have to go back to iTunes and instead of going to the album, go to the songs it previews right below it. The missing songs will have the "buy $???" Option. You need to select that option, and once you do it will say you have already purchased it and will ask if you want to download again for free. Say yes and your missing song(s) will download. Repeat step for all other missing songs individually.