How about...
Mair Caoimhe ("MIER" "KEE-va")
Mair is the Welsh form of
Mary. I don't know of any Irish "M" names which have the long "i" sound you want.
Caoimhe is Irish and contains the hard "c" and "uh" sounds. Between the two names, you've got the sounds to form the nickname,
Mica.
Hope that helps.