I'm going to make a "SWAG" (Scientific Wild Ass Guess) here. I know there are some Yaesu models that use more than one microphone, but you have to make a settings change in the menu when you change the mic. Perhaps this is also the case with your iCom.
What is the model number on the microphone that your radio came with? Does your user manual list any optional microphones?
Are you sure your radio is a non-H model? I'm not sure the non-H model was ever sold in the U.S. Based on Icom's past practice with model numbers, the only difference between the non-H model and the H model is the power output. I wouldn't think that would have any impact on the microphone. It's possible that there is something else wrong with the radio.
One user manual for the H model that I found online lists the HM-98 as the standard microphone and shows no other wired microphone as options. The Universal Radio website is my go-to source for info about radio options. They list no other microphone for the H model and they don't show a non-H model. Icom IC-2710H. Icom IC2710H transceiver 2710H