We can say the following about the part of speech that "to score" is in the sentence:
"To score", as it is used in the sentence, functions as an adverb.
Infinitive phrases functioning as adverbs
In the sentence we are analyzing here, "to score" is an infinitive phrase. As a matter of fact, the whole phrase would be "to score another run."
Infinitive phrases can function as three different parts of speech:
In this case, the phrase functions as an adverb, because it indicates why Brian went charging to home plate.
Learn more about infinitive phrases here:
https://brainly.com/question/1704062
