Another answer.

The oil alarm module checks oil level in the motor mounted tank, loosen the oil tank cap slightly, run the motor until the tank fills from the main tank so full that it begins to overflow, tighten the cap. The module also compares the signal from the oil pump rotation sensor to an ignition pulse from one cyl, often #4, when any difference it alarms.

Find the green wire on the module and follow it back to see for sure which cyl coil it connects to, spark test that cyl to ensure it is not an ignition problem.

Definitely add oil to the fuel just in case it is an oiling problem. The excess oil will only cause it to smoke a bit more than normal, rather that than risk hurting the motor.