A Koopa's Revenge and its sequel both have a fair share of hidden cheats to alter your playthrough of the game. This page contains the locations of them all. and trying to use cheats in the boss rush is not possible.
Press X + Y on the world map. Only works once.
A Koopa's Revenge 2
In 1-6, Mario's Villa, right outside the boss door, throw a Mini-Mario at the 2 blocks before the lip at the far right. Once it detonates, the blocks it rested on will be destroyed. Going down the new gap leads to a Metroid-styled item room with a Birdo statue (based on the Chozo statues from Super Metroid) holding the Space Jump.
In 3-4, My Name is Jonah, wall-jump up the hole in the ceiling above the door at the end of the first room to an alcove containing three 10-Coins. The 2 tiles to the left of the Silver Coin counter are coloured darker than the rest, and can be jumped through. This leads to another Metroid item room, with another Birdo statue.
With the Space Jump, holding a direction and jumping will result in the player character doing a spinning jump with the momentum from the ground. This can be repeated in midair forever.
In 2-A, Secret Shrooms, there is a gap between 2 Mushroom Platforms near the start of the vertical segment, specifically between the 5th and 6th platforms. Just off-screen is a hidden pipe. In this room, the player needs to use the Space Jump to slip between Ripper enemies, also from Metroid. At the top is yet another item room.
At the start of 4-5, Jolly Toy Factory, ride the third 16-ton weight to the top. Once it stops moving, press Up to enter a hidden door. After entering the cave, use Space Jump to go back through the hole you entered, then go right to an alcove with an elevator and 2 Roulette Blocks. Either bring a Spike Flower beforehand, or try to get one from the Roulette Blocks. After taking the elevator, detonate the lone Mini Mario on top of the 2 block furthest to the left. Following this, use a Spin Dash to break the blocks to the top-left, leading into the last item room. (Screw attack can also be used)
The Screw attack is used by holding a direction and jumping, similar to the Space Jump. As well, both cheats can stack.
In 1-5, D.O. Nut Bridge, go to the 4 Question Blocks next to the wooden wall. Above the third block is a Hidden Block containing a 1-Up. Hold Up on this block and jump to enter an invisible pipe. From there, make a running jump to an invisible platform. Continue left and Air Man's theme from Mega Man 2 will play. Continue further left and you'll reach a short segment where you have to jump across Goblins, complete with them releasing Petit Goblins. At the end of the screen, you'll find the icon for Rush Coil. However, you now have to go back the way you came, possibly using your new robot friend.
In 2-3, I Ain't Afraid o' No Ghosts, enter the first door from the start. After being returned to the start of the level, press Up where the door was to enter a hidden door. From this room, make a running jump to the right edge of the screen to move the camera to the right, revealing a cloudy background. Heading further right leads to another Goblin segment, albeit this one provides a vine to the end of the stage.
Rush can be called in by pressing the R key, regardless of key mapping. Once positioned, you can bounce on the springboard in his back. Unlike the other cheats, Rush cannot be disabled, likely due to only being summoned if you press a key.
God Mode is obtained by defeating Final Confrontation Mario. This cheat does exactly what it says on the tin. You can phase through enemies, bounce out of pits, and interact with lava and poison as if you had the Bone Flower.
Power-up cycling is unlocked alongside with God Mode. Cycling power-ups is done by pressing the backspace key. This can also be used when God Mode is turned off
Enter the Sega Genesis Mortal Kombat blood code, ABACABB on the cheat menu. This code can be found, sprawled in blood, in the background of 2-3 by entering the second door while using the Boo/Bone Flower during the night. It is not the same as in the first game.