Great, I can see it now. It’s only a minor specificity issue. Because it’s printed over the plugin CSS, it’s overwritted by the default CSS. Please use this modified version with higher specificity, it will work:
div#ajaxsearchpro2_1 .probox .proinput input.orig {
background-image: url("https://rioolspot.nl/wp-content/uploads/2024/06/cropped-rioolspot-favicon-32x32.jpeg") !important;
background-position: left center !important;
background-repeat: no-repeat !important;
padding-left: 40px !important;
}