What is the ruling on praying in a mosque which belongs to the people of innovation and how do I pray if there is no mosque upon the Sunnah in the city, and it is correct for me to pray with my wife in my house?

Shaykh Rabīʿ ibn Hādī al-Madkhalī:

No. If there is a mosque then pray in it even if it is with the people of innovation. ʿAbdullāh ibn ʿUmar prayed behind an individual from the Khawārij and this (narration) was used as proof by al-Bukhārī. (Praying in) congregation is a great affair which is not to be replaced for something like this.

As for the people of innovation, then from them are those who are hidden (in their innovation), thus if his innovation is not evident then pray behind him. However, if his innovation becomes apparent to you then explain the truth and clarify (the innovation) to him. If his innovation is of disbelief and he is insistent upon it, then do not pray behind him after the evidence of disbelief has been established upon him – meaning he commits an innovation of disbelief (i.e calling upon other than Allāh ﷻ, slaughtering for other than Allāh ﷻ, seeking rain from other than Allāh or distorting the attributes of Allāh ).

If he commits innovations of disbelief and you have debated with him, clarified the truth to him from the speech of Allāh ﷻ, the speech of his Messenger and the speech of the Salaf, yet he is insistent (upon his innovation) and stubborn (against the truth) then do not pray behind him. As for if his innovations if they are hidden and are not innovations which take someone outside of the fold of Islām then pray behind him.