Paul,
They did not. The Illinois unit in that fight was the 16th Illinois supported by 4 companies of the 3rd Iowa and 1 company of the Hannibal Home Guard plus one six pounder and a few rounds of six pound solid shot. The whole fight was on the order of a Chinese fire drill in my opinion. Kind of on the job training. Eventually Grant arrived with the 21st Illinois and Logan came with his regiment they number of which eludes this tired old brain at the moment but that was later in the fight. The good thing was that they wasted a lot of ammo and did not hit much. That goes for both sides. The Feds did not want to waste what cannon ammo they had and the Rebs only had a few rounds of the right ammo for thier cannon that had wheels on it and the other one they only had the tube so they would have to rest it on something to fire it. Anyway, they did try to knock down the Seminary building where the Feds were but that only loosened a few bricks. Kind of seems like a Keystone thing to me. :)
As for the 23rd or the Illinois Irish Brigade, they were mustered in about two weeks before the "Battle at Monroe City". THey moved to Quincy on July 14. That would have been after the "hostilities" in Monroe County for the most part. On the 21st they moved to Jefferson City under Col. Jeff Davis. They did do some reconing in the area around Jefferson City in August at which time they came under over all command of General Grant. In September they marched to the seige of Lexington, Mo. While there they did engage General Price and the Missouri State Guards. After that they went to Camp Douglas as guards until June of 1862 when they went to Harpers Ferry and after that they stayed in the Virginias until the end of the unpleasantness.
FC