El cocinero sirve un plato nuevo en la cocina.

In Spanish, when you refer to a specific person’s role or profession (except after ser), you normally include the definite article.
• El cocinero means “the cook” that both speaker and listener know about.
• Omitting the article (“Cocinero sirve…”) sounds like a newspaper headline or is simply unnatural in everyday speech.

Sirve is the third‐person singular present indicative of servir (to serve). Servir is an e→i stem‐changing verb. The present tense forms are:
• yo sirvo
• tú sirves
• él/ella sirve
• nosotros servimos
• ellos sirven
In our sentence, él sirve = “he serves.”

Most descriptive adjectives follow the noun in Spanish: un plato nuevo (“a new dish”). You can say un nuevo plato, but that order:
• often emphasizes “another new dish” or contrasts with earlier ones
• can sound more literary or marked
In everyday speech, un plato nuevo is more neutral and common.

Adjectives must agree with the noun in gender and number.
• plato is masculine singular, so nuevo takes -o.
• For plural: platos nuevos.
If it were a feminine noun (e.g. copa nueva), you’d use nueva.

Here plato means “dish” (the food), not the physical plate. You infer that because:
• It’s paired with nuevo (“a new dish,” not a new plate).
• The context of servir implies serving food.

• en indicates location: “in the kitchen.”
• a la cocina would mean movement: “to the kitchen.”
• You cannot drop the article; en cocina is ungrammatical here. You always say en la cocina.

No.
• servir means to bring or provide food to someone (serve a dish).
• poner la mesa means to set or lay out the table (plates, silverware, etc.).
You wouldn’t say pone un plato nuevo for “serves a new dish,” because that suggests “he literally places a new dish down,” not “he serves it to diners.”