It's natural to want to crawl into a hole and pretend no one is knocking on your door (not literally) when you are behind on a deadline or struggling with something you promised. But don't!! It doesn't make the problem go away, it makes the client/customer resentful and mistrustful.
The best thing you can do when your client has a problem that you are involved in is to face it head on. Call them, email them, text them or show up on their doorstep and let them know your desire to work with them to find a solution. Let them know you aren't trying to be lazy or ignoring them. We all make mistakes and as long as you accept responsibility and do the right thing, they most likely appreciate your candor and feel a kindred spirit with you.