Leaflet. Icon marker follow to mouse

Question

i want create following icon marker to mouse when i choose place (gif)

How do i write this

JS

let objectType = document.getElementById('cursor');
document.onmousemove = function(event){
    objectType.style.left = event.clientX+'px';
    objectType.style.top = event.clientY+'px';
}

HTML + CSS

<img src="/public/images/icon/obj.png" id="cursor" class="cursor"/>

.cursor {
    position: fixed;
    visibility: hidden;
}

But this does not allow you to click on the map and it is superimposed on the interface

How to do it better?