Entity Framework include children that grand children are null

Seem like a simple one, but not coming up with the right thing. All that I need to do is get a count of child items that have it's own child null. Here is what I have been working with:

var data = await _context.award.Include(a => a.expenses.Select(p => p.invoice == null)).ToListAsync();

I have also tried other combinations here with no luck. The error I get is

InvalidOperationException: The property expression 'a => {from expense p in [a].expenses select ([p].invoice == null)}' is not valid. The expression should represent a property access: 't => t.MyProperty'.

I change it to match and it just triggers a new error.

I just want to get a list of award with it's list of expenses listed (fine with just the .ID if that influences the solution) where the invoice parent object is not set and is null.

UPDATE requested models

public class invoice
{
    [DatabaseGenerated(DatabaseGeneratedOption.Identity)]
    public int ID { get; set; }

    [ForeignKey("INV_NUM_ForeignKey")]
    public invoice_number fin_invoice_number { get; set; }
}

public class invoice_number
{
    [DatabaseGenerated(DatabaseGeneratedOption.Identity)]
    public int ID { get; set; }

    public int number { get; set; }

    public invoice invoice { get; set; }

    public string display { get { return string.Format("sps-{0}", (new String('0', 6) + number.ToString())).Substring(number.ToString().Count()-7, number.ToString().Count()); } }
}