I need to add a site title as a text below the logo for a site. The title should be
<h2> tag. The theme allows either image or text, but not both.
You will have to customize the
header.php of your theme (from Appearancece > Editor), I recommend you create a child theme first and then find #logo within the header.php and add the title below the logo.
I cannot show you exactly how since I don't know the
header.php of your theme, but after you find the where the logo is generated #logo you will know where to place your title.