Should they pray Taraweeh in the mosque even though the people there insist on some innovations?
In the answer to question number 50718, we stated that what some people do of reciting dhikr in unison after every two rak’ahs of Taraweeh is an innovation which was not done by the Messenger (blessings and peace of Allah be upon him) or by any of his Companions.
You have done well to teach the people the Sunnah and denounce innovation. What they should have done is return to the teaching of the Prophet (blessings and peace of Allah be upon him), which is the best and most perfect of guidance.
You should continue to pray with them and call them to the Sunnah, and you will find, with Allaah’s help, some who accept the truth and are keen to follow it, even if others reject it. “But if these disbelieve therein (the Book, Al‑Hukm and Prophethood), then, indeed We have entrusted it to a people (such as the Companions of Prophet Muhammad صلى الله عليه وسلم) who are not disbelievers therein” [al-An’aam 6:89].
But if you find that the people in the mosque are stubborn and arrogant and reject the truth, then if you can pray in another mosque where the people are keen to follow the Sunnah and to avoid innovation, that is better. But if that is not possible, then your praying in this mosque is good, and you have done what is required of you and perhaps they will be guided. “(The preachers) said: ‘In order to be free from guilt before your Lord (Allaah), and perhaps they may fear Allaah’” [al-A’raaf 7:164].
If the followers of the Sunnah were to forsake the mosque because of such innovations, that would be a cause of innovation increasing and becoming more widespread, and it would lead to the Sunnah being erased and dying out. So you have to pray in the mosque and explain the Sunnah to the people, adhering in that to the advice of the Prophet (blessings and peace of Allah be upon him) to be kind and gentle to the one who is ignorant.
May Allah help you to do all that is good.
And Allah knows best.