Most likely the noise from the mains which will be imposed on the supply to the microphone, which likely does not have the best PSRR as normally batteries are low noise. Not much you can do about that using a plain wall wart, which has capacitance from input mains side to the output side. Grounded one will also introduce mains noise from the ground onto the audio as hum, crackles and other noise, especially if you have anybody in the neighbourhood using a power line internet connection.
You probably are best off using rechargeable cells to power it, or a small external case with a larger pack of cells, or a 2s lithium pack and balancer, with external charger you plug in when not in use. The microphone will work with that, and still will be portable and last a long time per charge, plus you will not be tied to as socket outlet.