As Hot Licks points out, the asterisk is commonly used to denote a footnote.
The use was extended colloquially to denote that something took place under unusual circumstances. Its use in this manner dates from 1961, when American baseball player Roger Maris exceeded the record for number of home runs hit in a single season. The record had previously been held by Babe Ruth. Maris's record was noted with an asterisk because it was accomplished under different circumstances -- the season had been extended by eight games.
As described in the article "Roger Maris: Still 61* after all these years" in NewsDay:
After 55 years, The Asterisk That Never Existed continues to haunt the legend and family of Roger Maris, the first player to hit more home runs than Babe Ruth in a single season.
In 1961, the American League expanded its schedule from 154 games to the current 162. As Roger Maris closed in on Babe Ruth’s single-season home run record of 60, commisioner Ford Frick declared that Ruth’s record would stand if not broken by the Yankees’ 154th game and should Maris break the mark or tie Ruth after that, the records would be treated separately.
That differentiation commonly became known as The Asterisk.
In the case of the quote in the OP, they want a nomination with a clean sheet, with no reason for anyone to stand up and declare that the nominee obtained his position by unfair means.