An ngram shows that “le ventre vide” is much more frequent in the written language; that’s in keeping with the entry “le ventre creux” in the TLFi; it’s a synonym of “le ventre vide”; those two phrases are used mostly with the verb “avoir” or without a verb; however, I know from memory that other verbs and verbal expressions can be used: “se sentir le ventre vide”, “partir le ventre vide”, “aller travailler le ventre vide”, “passer la journée le ventre vide”; it’s understandable that “avec le ventre vide” appears nowhere in the entry, as it is rare in print. As both forms are correct it is to be surmised that the more literary one (whithout avec), should be less common in the spoken language and more elegant, and that the other form tends to be used in the spoken language more than in the written language in virtue of the added explanation it provides, the word “avec” making more explicit for locutors and listeners the relation of possessor to thing possessed; I give credit to that understanding insofar as the oral users of a language are generally not so well versed in the literary aspect of their tongue. Therefore, not knowing if one should, in one’s personal speech and as regards such matters, abide by the wise enough rule that we should speak to each according to his or her understanding, I conclude that both forms can be used in the spoken language, without changing the meaning, and that the form without “avec” will be the mark of a somewhat more elegant elocution, while the form with the preposition remains quite acceptable.

Use ”avec” anytime you want to emphasize your sentence/expression, intensify your ideas/the tone of your voice, or when you want to drag people’s attention to a particular thing/idea. Use it to modify a verb, an adjective, a sentence…or another adverb. And you’re always free not to use it at all and your sentence will still be correct, meaningful both grammatically and contextually.

1- On ne peut pas se concentrer le ventre vide.
(Normal sentence with no emphasis on ”le ventre vide”)

2- On ne peut pas se concentrer ”avec” le ventre vide.
(Same idea, but this time the ”avec” adverb is used to put a particular emphasis or accent on ”le ventre vide”.)

Here are some of the reasons:
1- ”Avec” is an adverb, that means, it has no gender and is uncountable.
2- An adverb brings/gives ”precision” into/ to a phrase
3- The use of an adverb in a sentence is not obligatory or necessarily. A sentence will still keep the same meaning with or without an adverb within it (except the sentence will miss it emphasis or ”spice” without an adverb.)
4- An adverb can modifier or precise the meaning of a verb, a ”qualifier”/ ”qualification” adjective, a whole sentence or of another adverb.

Here it is again in French :

L’adverbe est un mot invariable, il ne possède ni genre, ni nombre. Il apporte une précision dans la phrase mais n’est pas obligatoire.
L’adverbe peut modifier ou préciser le sens d’un verbe, d’un adjectif, de toute une phrase ou d’un autre adverbe….and so forth…

Knowing this kind of things will help you understand the use and roles of certain French words or grammar rules.