In the poem "Desert Places," what multiple denotations of the word "benighted" are functional in the poem?
The "denotation" of a word is the actual, dictionary definition. In the same vein, the "connotation" is the suggestion of the meaning of a word apart from what it explicitly means. By definition, benighted has two denotations: "overtaken by darkness or night" and "existing in a state of intellectual, moral and social darkness."
Frost uses "benighted" in the third stanza of the poem:
And lonely as it is that loneliness
Will be more lonely ere it will be less--
A blanker whiteness of benighted snow
With no expression, nothing to express.
In this case, the word denotes the first definition in a literal way. The poem takes place at night, so the snow is "benighted." However, the sense that you get from the poem (the tone of the poem) coincides more with the second meaning: "existing in a state of intellectual, moral and social darkness." The speaker is very lonely, stating in the fourth stanza that as physically alone as he is, he can't be more lonely than he is in his own mind:
They cannot scare me with their empty spaces
Between stars--on stars where no human race is.
I have it in me so much nearer home
To scare myself with my own desert places.
In this additional context, I'd say that the denotation of "benighted" is also the connotation.