Adding custom cart price with Ajax in wordpress

Try to change your add_custom_price_callback() as follows. Untested but should work.

function add_custom_price_callback() {
    $custom_price = intval($_POST['p_m']);
    $target_product_id = intval($_POST['s_o_v']);         

    foreach ( WC()->cart->get_cart() as $key=>$value ) {

        //Single product
        if ( $value['product_id'] == $target_product_id ) {
             $value['data']->set_price($custom_price);
        }
        //For variation
        if ( $value['variation_id'] == $target_product_id ) {
             $value['data']->set_price($custom_price);
        }
   }
   //to check if is work or not       
   wp_send_json(array($value['data']->get_price($custom_price)));   
}