This surname is a moderately common Ukrainian name and was formed from the Hebrew name MALACHI. After 988 A.D., every Slav, having been baptized, would undergo a ceremony, conducted by a priest, to receive a Christian name. Consequently, church names became the basis of the formation of surnames.
The surname MALASHENKO was formed, based on the male name MALACHI, which means " messenger of God" or, more precisely, "messenger of Elohim." Colloquially, in Slavic languages, that name had several forms (sort of, nicknames): Malaphei, Malahei, Malaha, Malash, Malasha, etc. The "MALASHA" version became the basis for the surname MALASHENKO.
Most likely, this surname was formed in the 14th century. Originally, the "-enko" suffix meant "son of." However, later on, that meaning was lost, and the suffix "-enko" remained only in surnames.
Please, note that, regardless of the way the surname is spelled, it is pronounced the same way for all the alternatives. The spellings with an "S" in the middle are still pronounced with the "SH" sound.