You should wait for other answers: but, with that meaning, it can be a name from North India
whose pronunciation changed on its way.
mukha is a Sanskrit word (I do not know its deeper roots; but it is attested in the vedic literature, and is still in use) meaning mouth or face, and was used in many contexts to mean principal. mukhya was the adjective derived from that, which in Hindi and related languages gives mukhiyA meaning headman of a village. This person is the head of the village government, and so fits your meaning.
The -u- is short as in English put, the -i- is short as in English hit. the -A is long as in English car. The first short vowels may have shifted to schwa in pronounciation, and English usually represents that as an -a- (as in about). The -iy- shifting to -ay- I don't understand, unless it represents the sound in English may, i.e. the -i- backed into -e-
Or it may be a different word in a different language :-)