$specialId = 123;
$customOptions[$specialId] = "Special instruction custom option";

// $imageId = 124;
// $url = 'http://example.com/url/to/image.png';
// $customOptoins[$imageId] = $url;

$params = [
    'product' => $product_id,     //product Id
    'qty'     => $qty,            //quantity of product
    'options' => $customOptions   //custom options
];

$this->_cart->addProduct(
    $_product,
    $params
);