What is the grammatical name of this expression 'that he would agree to run the state' and what is it function​

now, of course, the quality and the seasonability of these winter dreams varied, but the stuff of them remained.   they persuaded dexter several years later to pass up a business course at the state university—his father, prospering now, would have paid his way—for the precarious advantage of attending an older and more famous university in the east, where he was bothered by his scanty funds. but do not get the impression, because his winter dreams happened to be concerned at first with musings on the rich, that there was anything merely snobbish in the boy. he wanted not association with glittering things and glittering people—he wanted the glittering things themselves. often he reached out for the best without knowing why he wanted it—and sometimes he ran up against the mysterious denials and prohibitions in which life indulges.

there you go thats what we are missing

a simple sentence must have at least one dependent clause.