Well done Sean Chick! If I can expand on your correct answers:
Benjamin Prentiss “had benefit of a politician; not a very good one.” At start of the National Emergency, Illinois Governor Richard Yates appears to have supported Prentiss (sent him to Cairo to command the situation there.) After the death of Senator Stephen Douglas in June 1861, Orville H. Browning was parachuted into the empty Senate seat… and assumed “support” of fellow Quincy resident, Benjamin Prentiss.
McClernand was the “Leading Congressman from Illinois” and provided his own patronage; and managed to finagle patronage from President Lincoln (of the opposite political party.) McClernand kept Lincoln appraised of “the real story” regarding operations out West through frequent letters (much in the same way Ulysses Doubleday kept President-elect Lincoln appraised of the situation at Fort Sumter in 1860/61.) These out-of-official channel letters acted as counterpoint to Official Army reports.
John A. Logan appears to have benefited from patronage of John McClernand early on; and subsequently received “support” of President Lincoln (because Lincoln needed southern Illinois Democrats to remain loyal to the Union.) Over time, the self-actualized John Logan became his own patron.
John Fremont. As first Republican candidate for President (1856) Fremont established a connection with Lincoln after the November 1860 election. Non-West Point army officer and self-made millionaire (actual net worth disputed; but a wealthy man) with strong political connections in Missouri (married into the Democrat Benton Family) Fremont was sent by President Lincoln to Europe as Special Emissary, with mission “to buy up all the serviceable small arms and light artillery pieces available.” These weapons helped arm the North… and Fremont’s purchase took them out of the market for possible sale to the South. For his support, Fremont was anointed Major General and put in command of Department of the West, based at St. Louis.