When I went to our local aviary with my Nikon 5700 I discovered that by holding the lens to the wire it largely dissapeared ... though at first I got an error message because the wire was stopping the lens from focusing so I held it back clear until the camera had focus and then put front of camera to wire. If the bird is away from the wire and you have to use the zoom to get a tight framing that helps to hide the wire.
The other option is to clone out the wire in editing ... another labourious task for you Christina