Trying to add a variable to this array

I think I see two problems.

One certainly is a syntax error. You need to use the . operator to join the strings together.

private $_nodes = array("https://omdbapi.com/?apikey='" . $site_config['OMDBAPIKEY'] . "'&i=%s");

But the private implies this is initializing an object property. Using your array value there will cause

Fatal error: Constant expression contains invalid operations

If you want to assign it using that value, you can do it in the constructor instead.

private $_nodes;

function __construct()
{
    $this->_nodes = array("https://omdbapi.com/?apikey='". $site_config['OMDBAPIKEY'] . "'&i=%s");
}

($site_config is undefined in this example, but I don't really know where it comes from. You'll have to define it in the scope of the constructor somehow. Maybe pass it as a parameter.)

As Don't Panic's answer says, the first problem you have is that you have to concatenate (join) the strings together using the concatenation . operator.

As that answer says, you also can't define an object property with the value of a variable in a class definition.

You can, however, use a constant value. So you would need to first of all change $site_config to be a const rather than a variable:

const Site_Config = array(
    'OMDBAPIKEY' => 123456,
    /* other definitions */
);

class SomeUsefulClass {
    private $_nodes = array("https://omdbapi.com/?apikey='" . Site_Config['OMDBAPIKEY'] . "'&i=%s")
}

Whether this is a good idea or not is decidedly open to question...

The problem with your code is the concatenation of strings. In PHP you can do it in several ways

As of PHP 5.6.0 it's possibel to use scalar expression involving numeric and string literals and/or constants in context of a class constant.

It means that it's possible to do following

const KEY ='OMDBAPIKEY';
private $_nodes = array("https://omdbapi.com/?apikey={self::KEY}&i=%s");

But it's not the cleanest way. The better option is to add it to constructor f.e you can see different types of concatenation in following code

<?php
class Test {
    private $_nodes = [];

    public function __construct(array $site_config)
    {
        // simple concatenation with .
        $this->_nodes[] = "https://omdbapi.com/?apikey=". $site_config['OMDBAPIKEY'] ."&i=%s";
        // using sprintf my prefered way, noticed escaped %s with i=
        $this->_nodes[] = sprintf("https://omdbapi.com/?apikey=%s&i=%%s", $site_config['OMDBAPIKEY']);
        // using complex curly syntax
        $this->_nodes[] = "https://omdbapi.com/?apikey={$site_config['OMDBAPIKEY']}&i=%s";
    }

    public function printNodes()
    {
        var_dump($this->_nodes);
    }
}

$t = new Test(['OMDBAPIKEY' => 'abc']);
$t->printNodes();

It will print

array(3) {
  [0]=>
  string(36) "https://omdbapi.com/?apikey=abc&i=%s"
  [1]=>
  string(36) "https://omdbapi.com/?apikey=abc&i=%s"
  [2]=>
  string(36) "https://omdbapi.com/?apikey=abc&i=%s"
}