However, whatever your mistake, there’s a right way and a wrong way to handle it.
If you make a mistake, admit it. Do it right away.
Don’t hide – talk about it. Tell your client you understand why they’re upset. Listen to their complaint. Apologize. Then, tell them what you’re going to do to fix it.
The cover-up is always worse than the crime (see ref: Watergate, Union Carbide, Iran-Contra). Hiding makes it worse.
Add something to the original specs to make the client happy.
They’ll understand. They may even become a bigger fan!