The child_of parameter excepts an integer value of the specific term you need to get descendants from. This means that you need to get the ID from the On-going category term and then passing that ID to the child_of parameter.
To note, the child_of parameter returns all descendants of the term passed, this includes direct children and all children of those children etc. If you only need first level children of a specific term, then you need to pass the term id to the parameter parent.
If you do not have the exact id, you can use get_term_by to get the id and then pass that to child_of in get_terms
EXAMPLE
This example shows how to get the term ID if you only have the term name, like in your case On-going
$get_term_id = get_term_by( 'name', 'On-going', 'category' );
$terms = get_terms( 'category', array( 'child_of' => $get_term_id->term_id ) );
var_dump( $terms );