Most important thing is to solve the problem as soon as possible. If the issue is on a live site with lots of visitors, it should be escalated to the rest of the team, if it’s not easy to be solved. Furthermore, it’s important to notify the client that you are working on it currently.
The second important thing is to protect the team from creating a bad image because of the mistake, that made the client contact you on the first place. Spare the client from details about the people guilty or the root cause of the issue whenever possible, unless this information will help the client prevent the problem from happening again.
Issues caused by third parties are fine to be dissected in an email.
Don’t underestimate the importance of trust and respect that is build in your clients towards the team, as this is the main catalyst for your communication with them. If you loose their respect or trust, you’ll struggle to be of any help to them.