$(document).ready(function() {

var my_results = $("#my_table").DataTable({

    "processing": true,
    "serverSide": true,
    "ajax": {
        "url": "/my-internal-api-endpoint/",
        "data": function ( d ) {
            d.eventKey = $("#myInput option:selected").val();
            d.specKey = 0; // <- This is value which need to be changed
        }
    },
    "columns": [
        { "data": "date" },
        { "data": "event" }
    ],
    "fnInitComplete": function(oSettings, json) {
            $('#foo').val(json.foo);
            $('#bar').val(json.bar);
        }
    });

    $('.myClass').on('click', function(){

        var yoo = $(this).val();
        my_results.ajax.data({specKey: yoo}); // <- There is a problem
        my_results.ajax.reload();
    });
    $("#myInput").on('change', function() {
        my_results.ajax.reload();
    });
});