"Abe Lincoln stood at the White House Gate... When along came Lady Lizzie Tod, Wishing her lover good speed." Lincoln tries several times to take Richmond, and is foiled each time
Abraham Lincoln's wife was Mary Todd; this apparently become "Lizzie Tod[d]" in the ballad.
The song as collected by Davis appears to be a fragmentary account of the various Federal attempts to take Richmond in 1861-1862. The first attempt lasted only "one or two days," seemingly referring to McDowell's Bull Run campaign of 1861. This was followed by McClellan's Peninsular campaign of spring and summer 1862, seemingly not mentioned in the song.
The final stanza refers to Lincoln's "Burnside horse," which "stuck tight in the mire." Ambrose Burnside was in charge at the Battle of Fredericksburg, which may or may not be alluded to, and also commanded the "mud march," clearly the subject of the last line. - RBW