Escape current post from loop

The title of question is inaccurate. I think you don’t want to escape first post, instead you want to exclude current post. You can use the post__not_in argument in WP_Query.

$args = array( 's' => 'searchTerm' );

//Check if we are in a post of any type
if( is_singular() ) {
    $post = get_queried_object();
    $args['post__not_in'] = array( $post->ID )
}

$query = new WP_Query( $args );
if ($query->have_posts()){
    while ($query->have_posts()){
        $query->the_post();
        //echo the post
    }
    wp_reset_postdata();
}

UPDATE

As you are using the code in ajax request, you haven’t access to current post data in the ajax action hook. You need to pass the current post ID in the ajax request data.

For example (no tested. just writing here):

//Enqueue the scripts and localize js variables
add_action('wp_enqueue_scripts', 'cyb_scripts');
function cyb_scripts() {

    //register my-script
    wp_register_script( 'my-script', '/url/to/my-sript.js', array('jquery') );

    //enqueue my-sript and dependencies
    wp_enqueue_script('jquery');
    wp_enqueue_script('my-script');

    //Localize script data to use in my-script. Set here the post ID to exlude
    $exclude_post = 0;
    if( is_singurlar() ) {
        $current_post = get_queried_object();
        $exclude_post = $current_post->ID;
    }
    $scriptData = array(
                      'ajaxurl' => admin_url('admin-ajax.php');
                      'exclude_post' => $exclude_post
                     );
     wp_localize_script('my-script', 'my_script_data', $scriptData);

}

//Ajax action hooks
add_action('wp_ajax_nopriv_process_ajax', 'cyb_process_ajax');
add_action('wp_ajax_process_ajax', 'cyb_process_ajax');
function cyb_process_ajax(){

    $args = array( 's' => 'searchTerm' );

    //Check for post to exlude
    if( isset($_GET['exclude_post']) ) {
        $args['post__not_in'] = array( intval( $_GET['exclude_post'] ) );
    }

     $query = new WP_Query( $args );
     //....
     wp_reset_postdata();

}

The javascript:

jQuery(document).ready(function($){

    $.ajax({

        type: "GET",
        url: my_script_data.ajaxurl,
        data: {
            action: "process_ajax",
            exclude_post: my_script_data.exclude_post
        }
    })
    .done( function( response ) {
        //....
    })
    .fail( function( response ){
        //....
    });
});