in functions

// get from session your URL variable and add it to item
add_filter('woocommerce_get_cart_item_from_session', 'cart_item_from_session', 10, 3);
function cart_item_from_session( $data, $values, $key ) {
    $data['text'] = isset( $values['text'] ) ? $values['text'] : '';
    return $data;
}

// this one does the same as woocommerce_update_cart_action() in plugins\woocommerce\woocommerce-functions.php
// but with your URL variable
// this might not be the best way but it works
add_action( 'wp_loaded', 'update_cart_action', 99);
function update_cart_action() {
    global $woocommerce;
    if ( ( ! empty( $_POST['update_cart'] ) || ! empty( $_POST['proceed'] ) )) {
        $cart_totals = isset( $_POST['cart'] ) ? $_POST['cart'] : '';
        if ( sizeof( $woocommerce->cart->get_cart() ) > 0 ) {
            foreach ( $woocommerce->cart->get_cart() as $cart_item_key => $values ) {
                if ( isset( $cart_totals[ $cart_item_key ]['text'] ) ) {
                    $woocommerce->cart->cart_contents[ $cart_item_key ]['text'] = $cart_totals[ $cart_item_key ]['text'];
                }
            }
        }
    }
}

// this is in Order summary. It show Url variable under product name. Same place where Variations are shown.
add_filter( 'woocommerce_get_item_data', 'item_data', 10, 2 );
function item_data( $data, $cart_item ) {
    global $post;
    if ( isset( $cart_item['url'] ) && is_cart()) {
        $data['text'] = array('name' => 'Is jouw bestelling een cadeau? Vul hieronder dan een tekst in, dan doen wij een kaartje met jouw tekst in de doos.', 'value' => $cart_item['text']);
    } else {
        $data['text'] = array('name' => 'uw persoonlijk bericht', 'value' => $cart_item['text']);
    }
    return $data;
}

// this adds Url as meta in Order for item
add_action ('woocommerce_add_order_item_meta', 'add_item_meta', 10, 2);
function add_item_meta( $item_id, $values ) {
    woocommerce_add_order_item_meta( $item_id, 'Cadeau?', $values['text'] );
}

in cart.php

					<td class="product-name">
						<?php
							if ( ! $_product->is_visible() ) {
								echo apply_filters( 'woocommerce_cart_item_name', $_product->get_title(), $cart_item, $cart_item_key ) . '&nbsp;';
							} else {
								echo apply_filters( 'woocommerce_cart_item_name', sprintf( '<a href="%s">%s </a>', esc_url( $_product->get_permalink( $cart_item ) ), $_product->get_title() ), $cart_item, $cart_item_key );
							}

							// Meta data
							echo WC()->cart->get_item_data( $cart_item );

							// Backorder notification
							if ( $_product->backorders_require_notification() && $_product->is_on_backorder( $cart_item['quantity'] ) ) {
								echo '<p class="backorder_notification">' . esc_html__( 'Available on backorder', 'woocommerce' ) . '</p>';
							}
						?>

                                                    <?php

                                                        $html = sprintf( '<div class="url"><textarea rows="4" name="cart[%s]" value="%s" title="Url" class="input-text url text" ></textarea></div>', $cart_item_key, esc_attr( $values['text'] ) );
                                                        echo $html;
                                                    ?>
					</td>