Is it possible to change any of the HTML/URL returned from the_category()

I’m not sure why you need this or what is your exact use case here, but, in short, the_category() simply echos the returned value from get_the_category_list() which uses get_category_link() which uses get_term_link() to get the URL of the category page.

This is where you should look if you need to change the URL of link linking to the category page. In get_term_link(), there is a term_link filter that you can use to alter the URL returned for a specific term/category

return apply_filters( 'term_link', $termlink, $term, $taxonomy );

So, if you need to change the URL of term ID 1 for the category taxonomy, you can try something like the following piece of code (NOTE: The code is untested)

add_filter( 'term_link', function ( $termlink, $term, $taxonomy )
{
    // Check if we have the correct term and taxonomy, if not, bail early
    if (    $term->term_id != 1 // Check if current term is 1 or not
         && $taxonomy != 'category' // Check if the current term is from the taxonomy category
    )
        return $termlink;

    // If we came to this point, we have the desired term and taxonomy, so lets alter the URL
    // Here should be the code to construct your new URL, you need to work on this
    $termlink = 'VALUE_OF_NEW_URL'; // This will be the new URL our term to link to its term page

    return $termlink;
}, 10, 3 );