McClure and Lynette often oppose each other, as McClure is more rational and less ideological than the first citizen. Around 2241, some citizens have petitioned McClure and the council to override some of the first citizen's decisions, which created quite a stir.
The following is based on Fallout 2 cut content and has not been confirmed by canon sources.
A cut ending would have it so that in the elections that occurred after the destruction of the remnants of the United States government, Senior Council Member McClure was to be appointed First Citizen. Under his rule Vault City was to join the NCR. 
End of information based on Fallout 2 cut content.
One of the threats the Chosen One can say to McClure about getting their [fake] Citizenship revoked after insulting him reads "Your powers are weak, old man. You are no match for me.", in reference to Darth Vader in Star Wars: A New Hope, in which he threatens Obi-Wan Kenobi during a lightsaber duel.
↑Fallout Bible 0:Lynette: "The Vault City that I helped establish was to outlast me and continue on for many more years. In the elections that occurred after the destruction of the remnants of the United States government, Senior Council Member McClure was appointed First Citizen and I retired to honorary council member status. With my new free time, I traveled south to NCR and met the NCR President. I was responsible for much of the legislation that followed in the years between NCR and our City."