get_the_modified_date should do what you’re looking for. I am guessing that perhaps you are struggling a bit with the sprintf syntax.

 $date = sprintf( '<span>' . __( 'Posted', 'theme' ) . ' <a href="%3$s"><time class="entry-date" datetime="%1$s">%2$s</time></a></span>',
    esc_attr( get_the_date( 'c' ) ),
    esc_html( time_ago() ),
    esc_html( get_day_link( get_the_date('Y'), get_the_date('m'), get_the_date('d') ) )
  );

Let’s break it down:

sprintf argument 1 – the string with placeholders

sprintf( '<span>' . __( 'Posted', 'theme' ) . ' <a href="%3$s"><time class="entry-date" datetime="%1$s">%2$s</time></a></span>',

This is the main string that will be built. It has 3 placeholders. However, they are in a non-traditional order.

1. The link (<a href=%3$s) has a placeholder for variable #3, a string.
2. The datetime attribute (datetime=%1$s), which is part of the HTML time element that makes the time machine-readable (useful for advanced features, and if the time element contains more than just the actual time), has a placeholder for variable #1, a string.
3. And finally the displayed time (>%2$s</time>) is looking for variable #2, a string.

sprintf arguments 2,3, and 4 – the values

Those 3 variables are provided in the following function arguments (again, not in the order they are used):

esc_attr( get_the_date( 'c' ) ),
esc_html( time_ago() ),
esc_html( get_day_link( get_the_date('Y'), get_the_date('m'), get_the_date('d') ) )

The first, which is feeding into that datetime= attribute, is getting the post’s published date and outputting it in the ‘c’ format, which is a standard date format that looks like 20019-03-27T15:19:21+00:00.

The second, which is the displayed time, is referencing a function I don’t recognized, called time_ago().

The third, which is being used for the link href=, is generating the link based off of the post’s published date.

So, to properly change the date and still use that function, we should adjust all 3 parameters.

$date = sprintf( '<span>' . __( 'Posted', 'theme' ) . ' <a href="%3$s"><time class="entry-date" datetime="%1$s">%2$s</time></a></span>',
    esc_attr( get_the_modified_date( 'c' ) ),
    esc_html( get_the_modified_date(),
    esc_url( get_day_link( get_the_modified_date('Y'), get_the_modified_date('m'), get_the_modified_date('d') ) )
  );

Note I changed the 3rd parameter to esc_url since it is a URL that is being generated, and that the display of the output date can be adjusted by passing in PHP date formatting options, such as get_the_modified_date( ‘m-d-Y’ )

Finally, all of this is getting stored in your $date variable as HTML, so make sure to actually echo it to the page before you’re done.