Can you tell me who is responsible for ensuring that the Azan in a mosque is called correctly, the Imam or the people running the mosque? If we have two imams shouldn't one do Azan instead of someone who cannot pronounce it properly?

All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.

The administration of the mosque, its Imam and its Muath-thin, are all responsible for the correctness of the Athaan and its soundness.

The Prophet said: “Each one of you is responsible for what's under his supervision.” [Al-Bukhari and Muslim]

So, the administration should appoint competent people that know the Sharee’ah rulings and apply them. Such people are responsible for this great job which is calling the Athaan and leading Muslims in prayer.

Ahmad reported that the Prophet said: “The Imam is the responsible and the Muath-thin is the trustworthy. May Allah guide the Imams and forgive the Muath-thins.”

In the narration of Ibn Maajah, the Prophet said: “The Imam is responsible. If he does well, then he will have the reward and so will they, but if he does bad, then that will be counted against him but not against them.”
Therefore, the Athaan should only be called by someone who can perform it perfectly.

Allah knows best.