Rid your Murray 21678A mower of any old gas that has been left from a previous season: Supplement old gas with a little fresh fuel, then start the Murray mower and let it idle to burn off all the old gas. Fill the tank completely with fresh fuel.
Unscrew the fuel tank cap and check to ensure it is not clogged. There is a small vent hole on the tank cap, and this must be unclogged to allow air to enter the tank so that the gas can flow to the carburetor. If you cannot blow through the cap, use a small pipe cleaner or straightened paper clip to force the clogged dirt and debris through the vent cap's hole. Wipe the cap with a soft, dry cloth to clean off any dislodged debris, then screw the cap back into place. This should help ensure that the Murray mower receives adequate fuel flow and narrow down the potential problems to various engine and engine-related issues.
Pop the engine hood if the mower's woes are not related to its fuel. Find the spark plug at the side of the engine assembly. Ensure that the thin black wire lying behind it is securely connected to the rear of the plug. Directly reinsert it if it is not.
Disconnect the plug wire from the Murray 21678A mower's spark plug gripping the cap firmly and pulling. Using an appropriate socket with a socket wrench, replace the spark plug entirely. Connect the plug wire to the new spark plug.
Look over the battery and ensure that both the black negative terminal wire and the red positive terminal wire are securely connected to the bolts on either of its sides and that they're on the right terminals. Reconnect them by placing the terminal end(s) over each respective bolt and twisting the bolts clockwise with a socket wrench.
Locate the solenoid between the battery and the mower's engine. The other end of the battery's red terminal wire stretches back to the solenoid. Ensure it is connected to the side bolt of the solenoid. Place the wire's end over the bolt and tighten it with the socket wrench.