I was on a call a few days ago, where the vendor was contentiously defending themselves because they hadn’t delivered the project on time, or on budget.
The call made me very uncomfortable.
If you’re wrong, say you’re wrong. Then fix it.
The worst thing you can do is hold firm, in the face of all evidence to the contrary, and claim that you are not wrong. Furthermore, when you start blaming your client, you know the relationship is on thin ice.
Do good work. Stand behind your work. When you’re wrong, admit it and fix it.