In your child theme’s style.css file, there is this rule:
.site-header .search-field {
background-color: tranparent;
background-image: url("images/search-icon.png");
background-position: 5px center;
background-repeat: no-repeat;
background-size: 24px 24px;
border-top-width: medium;
...
transition-delay: 0s, 0s;
width: 0px;
}
I would probably change the background-color value to white (or #fff).
For the magnifying glass search image, change the background-image url to:
background-image: url("https://marginalia.wendywainwright.com/wp-content/themes/twentythirteen/images/search-icon.png");
So that it’s pointing back to the icon in the parent theme. Or you can create your own images folder under your child theme, create your own search-icon.png file, and put it there.