Unless your child is very different from mine, I'm not sure giving a lecture is a great approach in this kind of situation. Lectures tend to turn in to what I call "Peanuts mode", referring to the classic American cartoon featuring children as the main characters, where the adults were never comprehensible - their speech is replaced by a trombone sound (in the TV versions). The child just tunes them out and gains nothing other than some added frustration.
I'm guessing from your post that you're the father, and you're frustrated that Mom isn't backing you up when you've had an interaction go poorly with your child. I've been there, as have I think we all. (Heck, I was there yesterday!) The important thing here is that you and Mom are on the same page overall: that is how you'll help your child the best. Whether she does the same thing as you in each instance is not nearly as relevant, though it certainly can be frustrating when you're in the moment; but remember, you're the adult, not the child - sometimes it's necessary to yield a bit in order to get to the best outcome.
When I've had a bad interaction with one of my children, where I wasn't able to gain their cooperation in something, it tends to be challenging to recover from that. It's human nature to get stressed and be more aggressive. But outside of the moment, hopefully you can see that being aggressive isn't helping the situation: what you need is for the child to understand the social situation, to learn how to manage their emotions properly, and how to react in a way that is constructive rather than destructive. Adding stress to the situation doesn't help that: it makes it much less likely that the child will learn.
What does help, though, is having a second parent who isn't amped up, and doesn't have their emotions tied up in the situation. That person can defuse the child, help them manage their emotions, and bring them back to ground, where they can then process the situation logically, instead of emotionally, and handle it properly.
Does that mean Mom should just forget that the issue happened? Absolutely not. The point of this isn't just to calm the child down, but also to help them manage the situation. But that's not going to be accomplished by calming them down and then stressing them again by confronting them, right? What it will be helped by, is Mom calming the child down, and then talking with them about the situation - asking the child what they think happened, talking about what they can do to make it better, both now (apologize, clean up the spill, put away the laundry, whatever) and next time (if you're interrupting in the middle of an important part of a game, give them the language to explain to you what they need - if they need 1 more minute to complete to a save point, to clearly explain that and clearly indicate they will do the thing you ask once that's bee done, for example).
This is something that should work both ways, unless one parent has the overwhelming majority of interaction time with the children. My wife and I regularly switch off in these roles - I'll get frustrated dealing with something, she'll step in and handle it; or she'll get frustrated dealing with something, and I'll step in and handle it. It's important that it work both ways, in my opinion, because that's when you and she get the empathy and understanding to recognize what the other person is doing. If she's always intervening to defuse your situations, but you never step in to defuse the situation when she's stressed, she won't understand how you feel.
This took my wife and I years before we were entirely comfortable with the other stepping in and handling things differently. We still sometimes get frustrated when the other steps in, in fact, though much more rarely - it's not easy being in a position where you feel like it's important that the child behave in a particular way, and he's not, and you are emotionally invested in the situation, and then the other parent steps in and handles it differently. It makes you feel like you're not good enough of a parent, and it makes you feel like your feelings don't matter. But just remember, as a parent you need to check your ego at the door as best as you can; the child's development is what's most important here, and it's up to you and Mom to make that happen in the best way you can.
