Complex joins with SUM

Assuming you just want to add a calculated column showing the aggregated payment for each line item that was made within a date range, you could implement it in the form of a correlated subquery:

SELECT
  ... /* all your current columns */
, (
    SELECT
      SUM(pp.amount)
    FROM
      payment_parts AS pp
      INNER JOIN post_type_ar_payment AS ptap
        ON pp.payment_id = ptap.id AND pp.invoice_id = ptap.invoice_id
    WHERE
      ptap.payment_date BETWEEN '2018-08-01' AND '2018-08-31'
      AND pp.invoice_line_item_id = invoice_line_item.id
  ) AS payment
FROM
  ... /* the rest of the query */
;

Demo at SQL Fiddle: http://sqlfiddle.com/#!9/721ea5/3