What should we call a part of a sentence which is neither a word nor a phrase?

Solution 1:

Every linguist has this problem, especially in talking about syntax to non-linguists.
I only use the term phrase to refer to constituents; but there is syntax for non-constituents, too.

Conversational Deletion, for instance, chews away at the beginning of a sentence,
producing utterrances like these, which lack some initial sequence of predictable words:

• Gotta go now.
• See you next Tuesday.
• No need to get upset about it.
• Ever get to Toledo, look me up.
• Good thing we didn't run into anybody we know.
• Last person I expected to meet was Harry.

What gets deleted varies considerably, and isn't limited to one type of word or construction.
Instead, it's just the little words at the beginning that get deleted.

What to call that sequence?

I'd call it a string, which simply refers to any series of successive words. YMMV

Solution 2:

A phrase is:

a sequence of two or more words arranged in a grammatical construction and acting as a unit in a sentence.

(from dicitonary.reference.com)

"Two" here means the shortest possible unit in a sentence. All clauses are phrases; but not all phrases are clauses.

In music, two or more notes played in succession are a group; anything more than two can be a theme, a grand theme, a leitmotif, a melody, and so forth. All of those are groups. But you can't have a theme consisting of just two notes, or a melody of merely three.