French troops were deployed on Bangui's main roads and in several districts of the capital on Thursday after gunfire and clashes on Wednesday.
Christian anti-balaka militias attacked Chadian soldiers, whom they accuse of backing Muslim Séléka militias,on Wednesday, killing five with another dying from his wounds on Thursday morning.
There were also civilian casualties with a journalist reporting seeing three bodies in a northern neighbourhood where fighting took place on Christmas Day.
Hundreds of residents fled the PK12 area on Wednesday and gunfire was heard elsewhere.
The French have reoriented their efforts towards the most troubled districts, such and Gobongo and Bacongo, officials said.
In a joint statement published in the Paris Le Monde Monseigneur Dieudonné Nzapalainga and imam Omar Kobine Layama, the president of the Muslim community, call on the UN to deploy a peacekeeping force "with the utmost urgency".
France's 1,600 troops and the 4,000 of the African force, Misca, "cannot bear the burden alone", they argue, adding that, although "reconciliation and peace are still possible", the country "remains on the verge of a war with religious aspects" and there is a danger of "terrible reprisals against Muslims" for the atrocities carried out by Séléka fighters.
When it endorsed the French mission to the CAR, the UN Security Council agreed to extend a UN mandate to Misca when conditions allowed.
Nearly half the population "desperately need aid", they say, and about 40,000 people have taken refuge at Bangui airport where they are living without shelter or toilet facilities.