Okay here are some more pictures to show where it is. I will do my best to explain.
This is from sitting on my patio looking to the back fence. Where all those bricks are laying around, it used to be a retaining wall that we are taking out now. The red circle is where the manhole is.
This is just a little closer up
And this is looking towards the back wall and to the side a bit. This part of the house was an addition, and you can see how they dug out the dirt but never leveled it back. So we have to work on that as well. I tried to draw a line so you can see how much higher the manhole cover will be than the yard itself.
So no it isn't exactly in the middle of the yard, it is probably about a foot from the back fence. However, it is still going to be really weird sticking out there. I assume the city has access to it, but it was completely covered by roots and everything, so nobody has used it in years. I am going to call them on Monday and see what exactly it is for, and what we can do about it.
Try and ignore the mess that the backyard is currently, that is why we are working on it. I will go back and find some pictures of what it looked like when we first bought it, and you can see how awful it was when we started.