This may clarify the permissibility of praying in the mosques which are a bit off the qiblah though they are still within its general direction.
The Lajnah ad-Da’imah stated:
… أما القبلة فأمرها واسع كما قال صلى الله عليه وسلم: ما بين المشرق والمغرب قبلة بالنسبة للشمال والجنوب ما بين المشرق والمغرب قبلة ، وبالنسبة للمشرق والمغرب ما بين الشمال والجنوب قبلة ، لكن يشرع التحري الكامل للقبلة ، هذا هو الأفضل ، وإلا فلا يضر الميل اليسير بالنسبة لليمين والشمال ، ما يضر والحمد لله .
As for the Qiblah, then its affair is spacious [i.e. the ruling is very flexible] like what [Muhammad] Sallallaahu Alayhi WaSallam said: Whatever is between the East and the West is a Qiblah. For the North and the South, whatever is between the East and the West is a Qiblah. And for the East and the West, whatever is between the North and the South is a Qiblah. However, what is legislated is to completely strive to achieve [the exact direction of the qiblah], this is what is better. Although there is no harm in being off (in direction] by a mere mile to the right or to the left, there is no harm (by it) and all Praise is to Allaah.