Its either the case or your finger or the camera.
I had this issue with my 4s.
One issue was the case i had was reflecting light back across the lens.
Take case off and try. It went away.
I bored the opening of my case and fixed it.
But before that i had an issue similar and it was camera. If you look closely into the lens under good light, I had what was an internal plastic gasket that had slid down and was kinda "half moon" at edge of lens
It was really close to covering the lens. It wasn't visible in images without flash, but it picked up flash when used and caused this.
They replaced the camera assembly quickly onsite and free. I looked at work order and it wasn't expensive even if i had to pay.
This is when i noticed case was some of culprit.
Compared to my iPhone 5 the lens is much more flare prone.
Side by side the 4s picks up lens flare from side/above lighting way more than the 5.
Many people have downed Apple. Saying their claims to lens improvement is mediocre.
I'm a professional photographer and i can tell you it is a HUGE improvement.
These new coatings and materials have made huge improvements in contrast, color saturation, and sharpness.
Wish I had a pic of the lens before they fixed it.
But there is literally a black foam gasket with a hole that goes around lens behind back glass that had come unstuck and slid out of place nearly covering lens.
Good luck