This situation is most noticeable when a singer adds a syllable to a word like "Fuh-ree" instead of "free" or "Buh-rave" instead of "brave". It's not Melisma, which is intentional by the writer.
It is epenthesis and anaptyxis, but those are rather vague terms. Epenthesis, in its function, ordinarily makes things easier to say, by breaking up difficult clusters, but the sort of addition being asked about here has an opposite function—it makes something more emphatic and, incidentally, harder to say. So I'd describe it as a fortition; basically as a lengthening, for emphasis.
Resonant consonants, i.e., liquids, glides and nasals, can be prolonged for emphasis. "Brave" -> "Brrrrave!". "Beautiful" -> "beee-youtiful". And a lengthened resonant is interpreted as its own syllable, whose vowel will be the vowel congener of the original consonant, if there is a natural choice of such a vowel. The vowel counterpart of the [j] glide in "beautiful" is [i] -- palatal vowel goes with palatal glide, so that's easy enough. In other cases, the choice of vowel that turns up in the new syllable is somewhat less obvious.
As I recall, this account of these emphatic syllables in English was first suggested to me by my teacher David Stampe.
Actually, what you're speaking of is called reduplication, which is the opposite of haplology.
Haplology involves dropping a syllable that is identical or very similar to an adjacent one, i.e., someone pronounces the word "preventative" as "preventive."
Reduplication is the opposite and involves repeating a syllable that is either identical or very similar, i.e., someone pronounces the word "supportive" as "supportative."
Hope this helps!
I had a very southern customer on a phone call who said (I.E.) my emal is Hail johnson @ google.net. He was asked to confirm it was hail like a hailstorm, he said no...............Ale johnson @google.net. He was again Asked again to confirm, Okay it's like beer , Ale right, he said "no...the letter ale" "L" is what he ment. Its a strong drawn out southern thing. I guess it's the addition of a syllable. I live in the south and have most of my life. Many variations to the southern dialect