There will be many times in your account management career when you will need to apologise to your clients. Deadlines missed, briefs not followed correctly, misheard instructions and targets not reached will all require a degree of atonement.
As an agency account manager (and company representative), when you apologise you are saying sorry on behalf of both yourself and your team.
It can be a pride-bruiser to have to apologise for something that may or may not be your fault. Nevertheless, you are the one who will need to take responsibility for the fact that something went wrong, that you didn't prevent it happening, and that it's your job to make sure it doesn't happen again.
The key to apologising - with humility and with genuineness - is to look at the situation from your client's point of view. As Seth Godin writes:
Apologize. Not because it’s your fault, but because the incident cost other people time or money or upset them, and you’re sorry that they have to deal with that.
If you can detach yourself from all twinges of shame or embarrassment, and focus on your client and your business relationship, then apologising will become easier.
When you have a spare minute you could dust off your kneepads, do some bowing exercises and practise delivering the best, most heart-felt apology of your career. When the time comes (and it will), your practise will pay off!