Adding (and then removing) a span to a div causes Javascript to not fire when img in the same div is clicked. How can I fix this?

Question

I'm building a chess app as a summer project and it works by submitting jQuery requests when pieces or squares are clicked. When a piece is clicked the request will return the valid squares for that piece to move to and then it adds a dot on that square to indicate that it can be moved there (it removes the dot after a square or other piece is clicked). The problem is that if there is already a piece on that square, the dot is added and then when it is removed, when the piece is clicked it no longer fires the Javascript. When I refresh the page it works fine again. How can I fix this so that even when a dot is added and then removed, clicking the piece will fire the js correctly?

This is the js code that runs when a piece is clicked:

$("img.piece").on("click", function(){
if (parent != undefined && parent.attr("class")=="light_square"){
    parent.css("background-color","WhiteSmoke");
    $(".dot").remove();
} else if (parent != undefined && parent.attr("class")=="dark_square") {
    parent.css("background-color","Sienna");
    $(".dot").remove();
}
parent = $(this).parent();
sq_one = $(this).attr("id").slice(5,10);
parent.css("background-color", "Gold");
$.ajax({
    contentType: 'application/json;charset=UTF-8',
    type:'POST',
    url:"/moves",
    data:JSON.stringify({'sq_one':sq_one}),
    success: function(response) {
        for (x in response) {
            var id = "square("+response[x]+")";
            document.getElementById(id).innerHTML += ("<span class=\"dot\"></span>");
        }
    }
    });
});

$(".dot").remove(); is also used when a square is clicked.

The CSS for dots, squares, and pieces is as follows:

.dark_square {
    background-color: Sienna;
    width: 45px;
    height: 45px;
    text-align:center;
    position: relative;
}

.dot {
  height: 20px;
  width: 20px;
  background-color: #bbb;
  border-radius: 50%;
  display: inline-block;
  opacity: 0.5;
  top: 50%;
  left: 50%;
  position:absolute;
  transform: translate(-50%, -50%);
}

.piece {
    width: 42px;
    height: 42px;
}

Finally I'm using a table to do the board and the html looks like this:

<td id="square({{ 8-x }},{{ y+1 }})" class="light_square">
    <img id="piece({{ 8-x }},{{ y+1 }})" class="piece" src="/static/images/{{ img_dict[(8-x,y+1)] }}">
</td>