What is the term for repetition of an initial syllable in successive words?

Rhyme — Literary Devices

Rhyme is a popular literary device in which the repetition of the same or similar sounds occurs in two or more words, usually at the end of lines in poems or songs.

But it's not restricted to word endings. Rhymes can be of many types.

For example, Rhymer gives the following taxonomy:

1. End Rhymes (blue/shoe)
2. Last Syllable Rhymes (timber/harbor)
3. Double Rhymes (conviction/prediction)
4. Triple Rhymes (transportation/dissertation)
5. Beginning Rhymes (physics/fizzle)
6. First Syllable Rhymes (carrot/caring)

[Italicization of mentioned words added]

Your example of Carla/Carlucci is an example of First Syllable Rhyme (for example, carrot/caring). Rhymer explains:

Words with first syllable rhyme have the same sounds preceding the first syllable break. For example, if you enter the word explanation using this option, Rhymer retrieves a list of words with the sound ex (e.g., excavate, exhale, expert, and extra). Other examples of first syllable rhyme include:

• pantaloons/pantomimes

• highlight/hydrant

• tulip/twosome

[Italicization of mentioned words added]

Alternate names for first syllable rhyme might include:

• beginning syllable rhyme
• initial syllable rhyme
• head syllable rhyme

I know of no term from prosody for the repeated use of a syllable at the beginning of successive words, but I imagine that there must be one.

In absence of a term from prosody, you might call this a reduplicated syllable or a reduplicated initial (syllable).

Reduplication is a technical term from linguistics to describe syllable or root compounds like "papa" and "pee-pee" which compound the same syllable or root twice. A subtype of reduplication is rhyming reduplication, which encompasses phrases like

hokey-pokey, razzle-dazzle, super-duper, boogie-woogie, teenie-weenie, walkie-talkie, hoity-toity, wingding, ragtag, easy-peasy (See here)

Your example of

Carla Carlucci

seems akin to the rhyming reduplication examples, except that the reduplication happens on the initial syllable. This is why I suggest initial (syllable) reduplication.

Adnomination — Wikipedia

Adnomination is the repetition of words with the same root word.

"News is what somebody, somewhere wants to suppress; all the rest is advertising." — Journalistic saying