How to edit only a specific part of each layer in a gif without messing up the loop?

Do you have the layer document the gif was created with saved as a PSD. If you just have the animated GIF file and open it in Photoshop.  photoshop will create a layer for each frame  So you would need the edit and add the glow to each layer you want the  glow in the  its corresponding frame. Or you add a new top layer that is not visible in all frames. Add the glow effect to that layer and add the layer to frames the have the eyes in the correct location.   

If you have the original layered document. The Animation may mad been made with the use of tween there may be far fewer layers then frames.  You would only need to edit the layers with the eyes that would be one or more layers.