Translation:His mouth is full, but he is trying to speak.
This is better to be answered by someone else, but when I am searching "has a full mouth" I am only able to find sentences about teeth and similar.
FWIW, "his/her mouth is full" is the best rendering in English. "He has his mouth full" and "she has her mouth full" are also regularly used and are accepted. I see this is an example of a literal translation that doesn't work very well, and I don't think it would be used much (except when talking about teeth).
"He has a full mouth but he tries to speak." Means the same thing but syntactically different.
It is NEUTER (edited, as per comment below), but it exists only in the plural form ("pluralia tantum"). You can find the declension here: https://en.wiktionary.org/wiki/%C3%BAsta (click Declension if the table isn't visible). A brief description of singular-only and plural-only nouns is here: https://mluvtecesky.net/en/grammar/plural_singular.
"He has a mouthful but is trying to speak." We often use mouthful rather than full mouth in English.
In the context of food, "mouthful," is generally used with reference to what the "mouthful" consists of, so I'm not sure it works here. See the example sentences for definition 1 at https://en.oxforddictionaries.com/definition/mouthful, all of which refer to a "mouthful of X."
This reminds me of the mít hlad construction, which is not translated as "to have hunger," but rather as "to be hungry." Maybe this is similar, since the best option in English seems to be "X's mouth is full," rather than "X has a full mouth."