It is helpful to pinch your nose closed when you have to swallow something that tastes yucky, like cough syrup.
In this sentence, how should I understand the word "closed" is it adverb or adjective? I do not think "your nose (which is) closed..
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The word closed is an adjective, describing the state of your nose after pinching. The structure is called resultative:
"In linguistics, a resultative (abbreviated RES) is a form that expresses that something or someone has undergone a change in state as the result of the completion of an event. Resultatives appear as predicates of sentences, and are generally composed of a verb (denoting the event), a post-verbal noun phrase (denoting the entity that has undergone a change) and a so-called resultative phrase (denoting the state achieved as the result of the action named by the verb) which may be represented by an adjective, a prepositional phrase, or a particle, among others. For example, in the English sentence The man wiped the table clean, the adjective clean denotes the state achieved by the table as a result of the event described as the man wiped."
In your example, the verb is "pinched", the noun phrase is "your nose", and the resultative phrase (an adjective) is "closed".