While WK does store the ID of items that have been partially answered in jStorage, what I did was just check whether the words should be visible every time an answer is submitted. You should look into what WK stores in jStorage. Setting up listeners for when the values change is particularly useful for figuring out when to make your script do something.
Might not be the best way to do it, but it was quick and easy. I just added this to your script
// Fires every time an aswer is submitted because WK updates the questionCount value
$.jStorage.listenKeyChange('questionCount', handle_answer)
// Toggles a class on the #reviews element on and off depending on whether the vocab should be visible
function handle_answer() {
// The field containing the user's answer. WK adds a class "incorrect" on this when you answer something incorrectly
let answer_field = document.querySelector('#answer-form fieldset');
// WK stores information about the current item in jStorage with the key "currentItem"
let item = $.jStorage.get('currentItem');
// If the item is a kanji review and the user answered it incorrectly, then remove the "hidden_vocab_list" class from #reviews
// Otherwise, add it to #reviews
$('#reviews').toggleClass('hidden_vocab_list', !(item.kan && answer_field.className == 'incorrect'));
}
// Insert some CSS hiding the #vocabulary_list when the class "hidden_vocab_list" is present on #reviews
$('head').append('<style id="kanjivocabcss">.hidden_vocab_list #vocabulary_list {display: none;}</style>');