Other field in match

You first have to add your new sports type using the filter

add_filter( 'leaguemanager_sports', 'my_sports' );
function my_sports( $sports ) {
   $key = 'my_key';
   $sports[$key] = __( 'My Sport', 'leaguemanager' );
   return $sports;
}

Then, you can add custom fields to the matches admin table using the following actions:

$key = 'my_key';
add_action( 'matchtable_header_'.$key, 'My_displayMatchesHeader', 10, 0 );
add_action( 'matchtable_columns_'.$key, 'My_displayMatchesColumns' );

$key has to be a unique identifier for your sports type. The input fields must be named in the pattern name="custom[$match->id][your_field]". Have a look at the files in the sports/ subfolder that use those actions, e.g. sports/gymnastics.php.