How can I clear permanent codes?

If you are looking at the codes on a scan tool and it shows it as a "permanent" code, not a pending, current or history code the short answer is that you cannot erase them. A pending code is one that has been set but isn't calling for the CEL until it is set a second time in a certain number of drive or ignition cycles. Current are codes that have set and requested the CEL on and are active at the moment. Both of those can be cleared with a scan tool or similar. History codes are codes that have been set in the past and are no longer either of the first two and have been repaired but can still be accessed to see if a current code has been set before. Then there are the permanent codes. These are current codes that were set, and they cannot be reset by anything other than the module that set them running the same test sequence a set number of times without having a pending or current code show up. Once those drive cycles occur the system will erase the permanent code.

It won't allow you to clear them. No scan tool will. A permanent code can only be cleared by the vehicles PCM. Say you have a P0300 listed as a pending a current and a permanent code. You repair the vacuum leak and clear the pending and current codes and the light is out. The PCM starts counting drive cycles and engine starts. If the P0300 code doesn't return in 20 drive cycles the PCM itself erases the permanent code. No way for anyone else to erase a permanent code. The reason for that is because a lot of people will put cars up for sale that have issues, they erase the trouble codes, and the light stays off for the test drive. You buy the car and the next day the light comes on. Seller says, "Hey it's never been on with me" So now the astute buyer will grab a small code reader and plug it in as soon as they can. If it shows pending or permanent codes, they have a way to either negotiate on price or just walk away. It also helps in shops if a vehicle comes in where someone else erased the codes.
Regardless though, if it is a permanent code, it stays there until the vehicle itself erases it.
If it isn't a pending or current code, it isn't anything to worry about. In your case those codes take 40 complete warm up cycles to clear. That would be a cold start, driving around until the engine is fully up to temperature, completing a drive cycle and then parking it and repeating the same steps 39 more times, once every 24 hours or so. Then the codes will clear. Not worth doing that if you know the issue is repaired.

If the issue went away on its own or not doesn't matter. It set a code. That code stays in memory until it cycles enough and then it will be erased. Or when the problem happens again it will return as a pending or current code. Their scan tool likely reads ALL codes as well so it sees the pending and current like most of the low end tools but it can also read history and permanent codes as well. But none of those tools, even factory ones will erase those codes. The vehicle itself has to see 30 full drive cycles without the event happening again and then those codes will be erased.
Because it "got better" I would plan on needing to repair whatever it was that set the code now, as it will return. More than likely you have corrosion or a failing sensor that set the codes. Things like that rarely repair themselves, they just take a while before they fully fail.
Do you know what the codes were?

No brainwashing involved. If a weather change caused it to set codes and the weather changing back cleared them, the problem is still there. It cannot magically repair itself. The codes it set will still be there until the system erases them. That is how almost every vehicle built after 2014 or so works, they store the codes in a way that they cannot be erased with a scan tool so that a shop that gets the vehicle in at a later date can still see the history of the vehicle. You say erasing the code made it run better, that sounds like it could be a cam crank code and you reset the KAM. Without knowing the exact code, I can't be sure as to which system is showing the fault. However, it doesn't change the fact that the codes cannot be erased by using any external tools.