Dear {Client Name},
On behalf of my company, {Company}, I would like to offer our most sincere apologies for the mistakes made while handling your project. It was a gross oversight for which we accept full responsibility.
To remedy this situation, we have already fixed our original error and will be offering you a refund of {Amount of Refund}. We try our best not to make mistakes, but when they happen, we try our best to fix them!
I hope that this will make amends for the mistakes that were made. Rest assured that your unfortunate experience has helped us identify a major problem in our {source of problem}, which has now been fixed.
If you have any questions about this issue, please contact us at {phone/email}. And, on behalf of all of us, we are truly sorry!
Sincerely,
{Sender}