Try
gadget_url.value=''
addGadgetUrl.addEventListener('click', () => {_x000D_
gadget_url.value = '';_x000D_
});
_x000D_
<div>_x000D_
<p>URL</p>_x000D_
<input type="text" name="gadget_url" id="gadget_url" style="width: 350px;" class="input" value="some value" />_x000D_
<input type="button" id="addGadgetUrl" value="add gadget" />_x000D_
<br>_x000D_
<span id="error"></span>_x000D_
</div>
_x000D_
I don't know why so many downovotes (and no comments) - however (for future readers) don't think that this solution not work - It works with html provided in OP question and this is SHORTEST working solution - you can try it by yourself HERE