No. The word actually comes from mwana + mume (it can also be mwanamume, but very often, that "m" is dropped and some people also say mwanamme, dropping the "u").
On its own, mwana means "child" as in "offspring" (rather than a person who is not yet an adult, which is mtoto), but in compounds it basically means "person". Mume means "husband"; the plural is waume.
Mwana can basically be thought of as mw+ana. The "m" that appears in singular is generally mw before a vowel. (This doesn't happen in mume though: m+ume, probably because the "w" is very hard to hear before a "u".) So the "w" is part of the prefix, not the stem. The regular plural of mw+ana would be wa+ana except that very often, two vowels coming together shorten into one. "Aa" often just becomes "a" and "ai" often becomes "e". For example, mwizi "thief" is mw+izi and the plural is mwezi "thieves", which is kind of like mwa+izi but the "ai" has become "e".
The plural forms of the M-WA can look a bit irregular if the singular starts with mw, but it helps to know what the verb is related to. There are some general tendencies ... nouns derived from verbs generally keep both "a"s. For example, the verb ku+andiki means "to write" and the word "writer" is derived from it. In singular, it's mw+andishi and in plural, it keeps both "a"s and is wa+andishi. Nouns derived from proper nouns also generally keep both vowels separate. For example Mwafrika "African person" is pluralised to Waafrika (NOT Wafrika!) and Mwislamu "Muslim" is pluralised to Waislamu (NOT Weslamu!).
The only example I can think of where the stem of the word begins with w is mwindaji "hunter", which is derived from ku+winda "to hunt". The singular is NOT mw+indaji but m+windaji so the plural is wawindaji (wa+windaji) and NOT wendaji (wa+indaji*).
It might seem complicated, but any M-WA word that begins with M + CONSONANT in the singular is simple, and words beginning with mwana- are VERY common and you can simply remember that they simply lose the m in plural.