After you earn enough in Warlock Zone VI to buy a pass, the tanuki statue lets you enter Tengu Mountain.
Warlock Zone VI: Tengu Mountain
Go to the left and fight the monkeys that hop slowly toward you. Go left until you have to jump up to the ledge, then go left until you have to jump up to another ledge. To the left, hit the pot to get a lucky cat. From there, go to the right until you can jump up the ledges. At the uppermost ledge, jump to the ledge on the left.
Keep going left and you will see a flying Karasu Tengu. It will try to float in a circle around you, then it will stop, flap its wings more quickly, then dash in a line with its sword raised.
Jump to the top of the tower where you first saw a karasu tengu, and hit the pot to get a gold cat, but don't go left onto the bridge unless you want to do an optional miniboss fight. If you want to skip the fight, be warned that the route is more difficult. To skip the fight, go back down the tower, then go left when possible, carefully jumping across the platforms and avoiding karasu tengu and monkeys. You can get $50 from the short tower in the middle, and a heart at the base of the leftmost tower. It might not be possible to reach the top of the leftmost tower from here, but you can try. There is a lucky cat on top of the leftmost tower, but if you can't reach it from below, you have to go back to the right, then up the rightmost tower, then fight the miniboss fight to get it. If you just want to skip the miniboss fight, go left to the next area and skip to Tengu Mountain Area 2.
If you don't want to skip the miniboss fight, then from the top of the rightmost tall tower (where you got a gold cat), go left onto the bridge and keep going left until the fight begins.
Tengu Brothers
The brothers with red masks will hop, then point their katana forward at you if you are nearby, then will jump to the other side of the screen. Keep your distance from the red brothers when they stop. When one of the red mask brothers is defeated, a blue mask one will take his place. The blue mask brother will hop, then throw a sharp leaf across the screen, then will jump to the other side of the screen. Jump over the leaf.
After they are defeated, go to the left and hit the pot on the statue's head to get a lucky cat. Then go carefully downward. There will be many karasu tengu to deal with. You might not be able to reach the short tower to the right because of how many karasu tengu are flying around. If you do manage to reach that tower, you can get $50 from there.
Then go left and downward to find a pot that contains a heart. Then go up until you can jump left onto solid ground, and go left to reach the next area.
Tengu Mountain Area 2
This area has monkeys and crabs, as well as fish behind the waterfall. The fish can't hurt you unless you go behind the waterfall, which you can do by jumping onto moving stone platforms and letting them take you behind the water.
First, jump onto the moving stone platform, but don't let it take you behind the water. (If you went downward first, go to the right, stand near the moving platform, wait for the moving platform to go behind the water, then jump up and to the right to reach the ledge where you first entered this area. From there you can jump onto the moving platform.) From the moving platform, jump left and hit the pot to get a checkpoint elephant.
Then go back to the right and jump onto the moving platform, and this time let it take you behind the water. Avoid the fish or hit them to get rid of them, and jump left onto the platform where the pot was. Jump up onto the three stone platforms, then jump right onto the two platforms there, then jump up onto the two platforms there. Jump left, then make your way up, making sure to stay behind the water. At the top, jump to the right, where there are four platforms side-by-side. Get on the rightmost of these platforms and let it take you in front of the water, then jump right and hit the pot to get a gold cat. Don't exit to the right from here or you will end up back at the area that leads to the bottom of the waterfall.
Wait for the moving platform to stick out, then jump onto it and let it take you back behind the water. Jump up to the next platforms. Don't exit at the top right or you will end up back at the area that leads to the bottom of the waterfall. At the top left moving platform, use coins or a yoyo to kill the crab, if possible. Exit at the top left and you will reach the boss of this area.
Kabuki
The box on Kabuki's back will open and shoot flower petals in the air. You'll take damage if any of the petals hit you, but you can hit the petals with a weapon to destroy them and make a safe area to stand in. The lightning jutsu will destroy all the petals at once. Then Kabuki will disappear and the box will float side to side. You can hit the box with a weapon to damage it, but avoid standing beneath it, because Kabuki will reappear after a while, and you'll get stomped if you're under the box. If you push up against the side of the screen when the box is floating from side to side, Kabuki won't reappear, and the box will just keep floating, allowing you to hit it with weapons or coins or bombs.
After Kabuki's box is defeated, Kabuki will jump around in a Kabuki pose, and he will periodically stop and throw his wig like a boomerang. Avoid getting hit by Kabuki and his wig. Try to run beneath Kabuki when he jumps while he's in the middle of the screen, but the timing can be difficult. You can try crouching while going beneath him to reduce the chance that he'll hit you while he goes past. Kabuki will sometimes also stop and blow out gusts of wind that will hurt you if they hit you. Jump over them. Keep hitting Kabuki until he's defeated.
Afterward, there will be a cutscene.