Smoking (?) Gun in “The Palace Thief”

Reviewing Ethan Canin’s story for the test, your classmate had this question: “After Charles Ellerby comes to ask Hundert about…the gun he receives from Senator Bell, and Hundert…gets rid of [it]. The next morning it appears that everybody unanimously turns away from Hundert and chooses Charles Ellerby, which is the moment Hundert knows that he is losing [the race for headmaster to] him [Ellerby].
This makes me wonder about the connection between [Hundert] throwing away his gun and the election of the headmaster. Are there any relations between Hundert losing his gun and his position as the headmaster? Does the fact that Ellerby knows about his secret affect him as a candidate or does he tell his colleagues about this “bad rumor” of Hundert and make them change their minds? Or could it be that the gun was somehow the symbol of the headmaster position?”