The optimization process and recommendations:

1. Avoid Correlated Subqueries (query line: 13): A correlated subquery is a subquery that contains a reference (column: patientid) to a table that also appears in the outer query. Usually correlated queries can be rewritten with a join clause, which is the best practice. The database optimizer handles joins much better than correlated subqueries. Therefore, rephrasing the query with a join will allow the optimizer to use the most efficient execution plan for the query.
2. Create Optimal Indexes (modified query below): The recommended indexes are an integral part of this optimization effort and should be created before testing the execution duration of the optimized query.

Optimal indexes for this query:

ALTER TABLE `invoice` ADD INDEX `invoice_idx_patientid_invoicednumber` (`patientid`,`invoicednumber`);
ALTER TABLE `payment` ADD INDEX `payment_idx_patientid_paymentdate` (`patientid`,`paymentdate`);

The optimized query:

SELECT
        patient.patientid,
        patient.firstname,
        patient.lastname,
        patient.mobilephone,
        patient.email,
        Format(Coalesce((SELECT
            Sum(invoice.amount) - Sum((invoice.amount * (invoice.discount / 100))) 
        FROM
            invoice 
        WHERE
            invoice.patientid = patient.patientid 
            AND invoice.invoicednumber > 0) - (SELECT
            Sum(payment.amount) 
        FROM
            payment 
        WHERE
            payment.patientid = patient.patientid),
        0),
        0) AS answer,
        Date_format((SELECT
            Max(payment.paymentdate) 
        FROM
            payment 
        WHERE
            payment.patientid = patient.patientid),
        '%d-%m-%Y') AS lastpaymentdate 
    FROM
        patient 
    WHERE
        1