Change the background-position
property value to something like 365px 5px
.
Updated Example - (Not sure where you want it positioned - but now it's visible at least)
.right-inner-addon input {
height: 30px;
padding: 0px 10px;
background-image: url('http://www.paulund.co.uk/playground/demo/pretty_search_forms/images/search.png');
background-position: 365px 5px;
background-repeat: no-repeat;
}