This paper toy is Blockhead grande, essentially immobile stone statue who's only purpose in life is to block passages, based on the video game Okami, the paper model created by scarykurt. Blockheads claim never to attack first as it is brutish, their clan motto being "Take the beating, then dish it out! That is our clan's philosophy!".
There are four Blockheads altogether, one in Tsuta Ruins, Kusa Village, Oni Island, and Kamui. With them having 2, 4, 5, and 8 weak points respectively. Blockhead Grande is the leader of the clan and therefore has the most weak points. The only Blockhead that needs to be defeated to defeat the game is the Blockhead in Tsuta Ruins. The one in Oni Island looks like it needs to be defeated but you can bypass it by Cherry Bombing a crack in the wall and using Vine on the newly revealed Konohana Blossom.
Blockheads are not damaged by any normal attacks. The only way to defeat it is to headbutt it, and a cut scene will reveal its weak points in a specific order which need to be re-dotted in the correct order with the Celestial Brush. If you fail to do this properly Blockhead will cause a rock avalanche with it's staff, although this can easily be avoided. The location of the weak points will change with every headbutt but the amount of them will not. For defeating a Blockhead with a lot of points you should try to quickly mark the points on your TV screen with something removable. You can use plastic wrap and a highlighter, but just make sure you recall to hit the weak points in order.
For that last Blockhead, Blockhead Grande, a great way to defeat him is by using a video camera that can record and playback frame by frame. Just make sure to hold down the action to paint right after all the weak points are shown. Otherwise you'll have to rerecord the new order of weak points after headbutting Blockhead Grande. The Blockheads seem sorrowful when they are defeated as they were not able to keep their post. However they die with no regrets.