Il cuoco mescola la zuppa nella pentola con un mestolo di legno.

In Italian, a specific, singular masculine noun normally needs the definite article:

• il is used before most consonants.
• lo is reserved for words beginning with s+consonant, z, gn, ps, x, etc.
  Since cuoco starts with a plain c, we say il cuoco. Omitting the article (“cuoco mescola…”) would sound odd if you mean “the cook” in a definite sense.

mescola is the present indicative, third person singular of the verb mescolare (“to stir”):

• io mescolo
• tu mescoli
• lui/lei mescola
  Because the subject (il cuoco) is “he,” we use mescola to express “he stirs.”

Here la zuppa is the direct object (“the soup”):

• Italian generally uses the definite article with specific or known objects.
• If you say mescolare la zuppa, you mean you’re stirring that soup.
  Dropping it (mescolare zuppa) would sound like you’re speaking very generically or abstractly.

nella is a contraction of in + la:

• in means “in” or “inside.”
• la is the feminine singular definite article.
  So in + la pentola → nella pentola.
  You cannot use nel (in + il) because pentola is feminine.

con here is the instrumental preposition, showing the tool used to stir:

• “with a wooden ladle.”
  You could rephrase with usando
  • gerund for “using…”:
     usando un mestolo di legno
    But con
    • noun is more direct and very common.

un is the indefinite article for masculine singular nouns:

• It indicates one ladle, unspecified.
• If you said il mestolo, it’d mean “the ladle” (a specific one).
• Omitting the article (mestolo di legno) sounds more like a label or a recipe heading, not a natural sentence.

di marks material or composition when used as a noun complement:

• un mestolo di legno = a ladle made of wood.
  By contrast, in legno functions more like an adjective phrase:
• un mestolo in legno also means “a wooden ladle,” often describing its style or characteristic.
  Both are correct, but di legno literally emphasizes the material.