Using wp_handle_upload() to Direct Specific Path by Using $overrides

You need to specify a list of allowed mime types.

You could make it easy by just getting the allowed mime types like:

$file = $_FILES['the-file'];
$upload_file = wp_handle_upload($file, array(
    'test_form' => false,
    'mimes' => get_allowed_mime_types()
));

If you look at the codex for Default allowed mime types, you could manually specify which ever mime types you want in that format.

An example would be like this answer.